#6856467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#39725	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1629 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)(9,10,13,12)(11,14,16,15)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15);(3,8)(5,7)(9,10,13,12)(11,15,16,14)(17,18,21,24)(19,23,25,30)(20,28,27,22)(26,32,31,29);(2,6)(5,7)(17,19,21,25)(18,23,24,30)(20,26,27,31)(22,29,28,32);(9,12,13,10)(11,14,16,15)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6763663);