#6909328

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#29745	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,6)(3,7,8,5)(17,18)(19,27)(20,24)(21,25)(22,31)(23,26)(28,32)(29,30);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(19,28)(20,30)(24,29)(27,32);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,29,23,28,22,24)(18,20,31,32,26,30,25,27);(2,6)(5,7)(11,16)(13,15)(17,20,23,30)(18,24,26,29)(19,25,28,31)(21,32,22,27);(10,14)(13,15)(17,21,23,22)(18,25,26,31)(19,29,28,24)(20,27,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816524);