#6801372

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667652
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	#25720	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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)(17,18,22,25,21,23,20,26)(19,27,29,31,28,32,30,24);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(18,23)(19,28)(25,26)(29,30);(10,13)(14,15)(17,19)(18,24)(20,29)(21,28)(22,30)(23,31)(25,32)(26,27);(2,6)(5,7)(17,20,21,22)(18,25,23,26)(19,29,28,30)(24,32,31,27);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,20,21,22)(18,26,23,25)(19,29,28,30)(24,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6708568);