#6802839

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	#28325	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)(17,18,23,24,22,28,19,27)(20,30,29,25,21,26,31,32);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,22,23)(18,24,28,27)(20,31,21,29)(25,26,32,30);(3,8)(5,7)(10,13)(14,15)(17,20,22,21)(18,25,28,32)(19,29,23,31)(24,30,27,26);(2,6)(5,7)(17,21,22,20)(18,26,28,30)(19,29,23,31)(24,32,27,25);(9,11,12,16)(10,14,13,15)(17,19,22,23)(18,27,28,24)(20,29,21,31)(25,26,32,30)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=i,d^2=i,e^2=fhi,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, 6710035);