#6767482

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1535 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6674678);