#6627009

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1449 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,29)(21,25)(22,27)(23,26)(28,32)(30,31);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,23,30)(18,24,26,31)(20,25,32,27)(21,28,22,29);(9,11,13,12)(10,15,14,16)(17,20,21,31,23,32,22,24)(18,19,27,28,26,30,25,29);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(17,21,23,22)(18,25,26,27)(19,28,30,29)(20,24,32,31);(9,12,13,11)(10,16,14,15)(17,21,23,22)(18,25,26,27)(19,29,30,28)(20,31,32,24)

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

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, 6534205);