#6802834

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	#32913	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1612 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)(17,18,22,23,21,25,19,26)(20,24,29,30,28,31,27,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,21,22)(18,23,25,26)(20,27,28,29)(24,30,31,32);(3,8)(5,7)(10,13)(14,15)(18,23)(19,22)(24,30)(25,26)(27,29)(31,32);(2,6)(5,7)(17,20,21,28)(18,24,25,31)(19,27,22,29)(23,30,26,32);(9,11,12,16)(10,14,13,15)(20,28)(24,31)(27,29)(30,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,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, 6710030);