#6636844

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636836
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,27)(20,22)(24,31)(28,32)(29,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18)(19,28)(20,25)(21,23)(22,26)(24,30)(27,31)(29,32);(11,15)(13,16)(17,19,22,27,21,30,20,29)(18,24,26,32,23,28,25,31);(2,6)(3,8)(11,15)(13,16)(17,20,21,22)(18,25,23,26)(19,29,30,27)(24,31,28,32);(2,6)(5,7)(10,14)(13,16)(18,23)(24,28)(25,26)(31,32)

Polycyclic presentation:

2^9|c^2=g,d^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

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, 6544040);