#72318738

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7624085	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17320 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2219 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,6,5)(3,7,11,10)(4,12,14,8)(9,16,13,15)(17,18,23,29)(19,31,34,48)(20,36,22,39)(21,30,24,27)(25,42,44,32)(26,38,28,37)(33,47,40,45)(35,46,41,43);(1,3,5,10,6,11,2,7)(4,13,12,15,14,9,8,16)(17,19,21,35,23,34,24,41)(18,25,27,45,29,44,30,47)(20,33,38,42,22,40,37,32)(26,43,39,48,28,46,36,31);(1,4)(2,8)(3,9)(5,12)(6,14)(7,15)(10,16)(11,13)(19,32)(20,37)(21,24)(22,38)(25,31)(26,36)(27,30)(28,39)(33,41)(34,42)(35,40)(43,47)(44,48)(45,46);(1,3,2,7,6,11,5,10)(4,13,8,16,14,9,12,15)(17,20,24,37,23,22,21,38)(18,26,30,36,29,28,27,39)(19,33,41,32,34,40,35,42)(25,43,47,31,44,46,45,48);(4,14)(8,12)(9,13)(15,16)(19,34)(25,44)(31,48)(32,42)(33,40)(35,41)(43,46)(45,47)

Polycyclic presentation:

2^10|a^2=h,b^2=f,d^2=fh,f^2=h,g^2=ij,i^2=j,b^a=bg,c^a=chj,c^b=cghij,d^a=di,d^b=dhj,d^c=dhij,e^b=ej,e^c=ehj,f^a=fj,f^c=fj,g^a=gi,g^b=gij,g^c=gi,g^d=gj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318841, #72318898, #72319381
maximal normal subgroup of	#72318841, #72318898, #72319381

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);