#72322843

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	#7623114	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17336 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2322 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=h,f^2=h,g^2=i,i^2=j,b^a=bgij,c^a=chj,d^a=dghj,d^c=dh,e^c=ej,f^a=fj,g^b=gi,g^d=gij,i^b=ij,i^d=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322909, #72343989
maximal normal subgroup of	#72322909, #72343989

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.7);;
gap> SetPower(coll, 2, F.6);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);