#6463264

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1522 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370460);