#6637410

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636852
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1641 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,27)(20,28)(21,26)(22,25)(23,30)(24,29)(31,32);(2,5)(3,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,24,25,27)(20,30,31,26)(21,32,23,28);(11,15)(13,16)(17,20,23,27,22,31,21,24)(18,19,26,28,25,29,30,32);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(18,25)(20,31)(24,27)(26,30);(2,5)(6,7)(10,14)(13,16)(17,21,22,23)(18,26,25,30)(19,28,29,32)(20,24,31,27)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6544606);