#6548020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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)(17,18,20,25,21,24,30,23)(19,26,28,32,29,22,27,31);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,22,24,26)(20,28,30,27)(23,32,25,31);(10,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,29)(27,31)(28,32);(9,11)(10,14)(12,15)(13,16)(17,20,21,30)(18,23,24,25)(19,27,29,28)(22,31,26,32);(11,15)(14,16)(17,20,21,30)(18,23,24,25)(19,28,29,27)(22,32,26,31)

Polycyclic presentation:

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

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, 6455216);