#72319189

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	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×#442
prime radical	#7623097	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2395 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)(3,12)(4,14)(5,16)(6,8)(7,11)(9,13)(10,15)(17,18)(19,23)(20,25)(21,26)(22,24)(27,34)(28,38)(29,36)(30,37)(31,41)(32,39)(33,40)(35,43)(42,48)(44,46)(45,47);(1,3,7,6)(2,8,11,12)(4,5,13,15)(9,16,14,10)(17,19,20,30,21,28,31,42,22,29,32,27,33,43,45,44)(18,23,25,37,26,38,41,48,24,36,39,34,40,35,47,46);(1,4,6,15,7,13,3,5)(2,9,8,16,11,14,12,10)(19,27,29,30)(20,31)(21,33)(23,34,36,37)(25,41)(26,40)(28,42,43,44)(32,45)(35,46,38,48)(39,47);(1,5,3,13,7,15,6,4)(2,10,12,14,11,16,8,9)(18,24)(19,28,29,43)(20,32)(23,35,36,38)(26,40)(27,42,30,44)(31,45)(34,46,37,48);(19,29)(23,36)(27,30)(28,43)(34,37)(35,38)(42,44)(46,48)

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=fh,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=ch,c^b=cghi,d^a=dhj,d^b=dij,e^b=ej,f^b=fj,g^c=gi,g^d=gj,i^c=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319265, #72319510, #72319579, …
maximal normal subgroup of	#72319265, #72319510, #72319579, …

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