#621728

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 36 points:

(2,3)(15,19)(18,20)(21,22,26,27)(23,25,29,34)(24,31,32,28)(30,36,35,33);(6,9)(10,12)(13,14,17,16)(15,20,19,18)(22,27)(23,29)(24,32)(33,36);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,19)(14,18,16,20)(21,23)(22,25)(24,30)(26,29)(27,34)(28,33)(31,36)(32,35);(1,3,4,2)(13,14,17,16)(15,18,19,20)(21,24,26,32)(22,28,27,31)(23,30,29,35)(25,33,34,36);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,20,19,18)(21,25)(22,23)(24,33)(26,34)(27,29)(28,30)(31,35)(32,36)

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=g,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 528924);