#6757339

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#39469	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,25,22,19,21,26)(20,24,29,32,28,27,30,31);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(20,28)(24,27)(29,30)(31,32);(9,11,12,15)(10,14,13,16)(17,19,22,18)(20,27,28,24)(21,26,23,25)(29,32,30,31);(2,6)(5,7)(10,13)(14,16)(17,20,22,28)(18,24,19,27)(21,30,23,29)(25,32,26,31);(3,8)(5,7)(17,21,22,23)(18,25,19,26)(20,29,28,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6664535);