#6637404

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

1700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|c^2=g,d^2=f,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, 6544600);