#6440910

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348106);