#6754459

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6661655);