#6798580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6705776);