#6453441

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1514 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,22,21)(19,24,27,25)(20,29,31,23)(26,32,30,28);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,19,22,27)(18,24,21,25)(20,30,31,26)(23,32,29,28);(9,11,13,12)(10,15,16,14)(17,20)(18,23)(19,25)(21,29)(22,31)(24,27)(26,28)(30,32);(9,12,13,11)(10,15,16,14)(17,21,22,18)(19,26,27,30)(20,29,31,23)(24,32,25,28);(1,3,4,7)(2,6,5,8)(18,21)(20,31)(24,25)(26,30)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,e^d=ei,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, 6360637);