#620869

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

1894 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fg,d^2=h,e^2=f,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, 528065);