#6622956

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1589 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)(3,7)(4,6)(5,8)(10,14)(15,16)(17,18,21,24)(19,26,28,31)(20,23,29,25)(22,30,32,27);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,16,12,15)(17,18)(19,22)(20,25)(21,24)(23,29)(26,27)(28,32)(30,31);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,14,16)(17,19,20,27,21,28,29,30)(18,22,25,26,24,32,23,31);(1,4,5,3)(2,6,8,7)(11,12)(15,16)(17,20,21,29)(18,23,24,25)(19,27,28,30)(22,31,32,26);(9,12,13,11)(10,16,14,15)(19,28)(22,32)(26,31)(27,30)

Polycyclic presentation:

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

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, 6530152);