#6433838

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1730 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,29)(21,28)(22,30)(23,27)(25,26)(31,32);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,22)(18,24,27,30)(20,26,31,28)(21,29,25,32);(11,15)(13,16)(17,20,22,26,23,31,19,28)(18,25,24,29,27,21,30,32);(10,14)(13,16)(17,21)(18,26)(19,29)(20,30)(22,32)(23,25)(24,31)(27,28);(17,19,23,22)(18,24,27,30)(20,28,31,26)(21,32,25,29)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341034);