#6800318

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

1534 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)(19,30)(20,26)(21,32)(22,29)(23,28)(24,27)(25,31);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(17,19,23,31)(18,25,28,30)(20,22,21,24)(26,27,32,29);(3,8)(5,7)(10,13)(14,15)(17,20,23,21)(18,25,28,30)(19,22,31,24)(26,29,32,27);(2,6)(5,7)(17,21,23,20)(18,26,28,32)(19,22,31,24)(25,29,30,27);(9,11,12,16)(10,14,13,15)(17,22,23,24)(18,27,28,29)(19,20,31,21)(25,26,30,32)

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=i,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, 6707514);