#6856517

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(17,18)(19,26)(20,28)(21,25)(22,23)(24,30)(27,31)(29,32);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,19,21,22)(18,23,25,26)(20,29,30,27)(24,32,28,31);(3,8)(5,7)(9,10,13,12)(11,15,16,14)(18,23)(19,22)(24,31)(25,26)(27,29)(28,32);(2,6)(5,7)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32);(9,12,13,10)(11,14,16,15)(18,25)(20,30)(23,26)(27,29)

Polycyclic presentation:

2^9|a^2=hi,b^2=fhi,c^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,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, 6763713);