#6883330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#30859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,24)(19,21,26,23)(20,28,30,25)(27,29,32,31);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(18,24)(19,26)(25,28)(27,32);(11,16)(13,15)(17,19,23,18,22,26,21,24)(20,27,29,25,30,32,31,28);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20)(18,25)(19,27)(21,31)(22,30)(23,29)(24,28)(26,32);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,26,24,19)(20,29,30,31)(25,27,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6790526);