#6745720

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

2299 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,22)(19,23)(20,28)(25,27)(26,31)(30,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30);(11,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31);(2,6)(5,7)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32);(3,8)(5,7)(11,15)(13,16)(20,28)(24,29)(26,32)(30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6652916);