#690290

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2078 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)(7,11)(10,12)(13,14,16,18)(15,19,20,17)(21,22,26,30)(23,24,28,27)(25,29,34,36)(31,32,33,35);(6,9)(10,12)(21,23,26,28)(22,27,30,24)(25,33,34,31)(29,32,36,35);(5,6,8,9)(7,10,11,12)(13,15,16,20)(14,17,18,19)(21,23)(22,27)(24,30)(25,31)(26,28)(29,35)(32,36)(33,34);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,14)(15,17)(16,18)(19,20)(21,24,26,27)(22,28,30,23)(25,32,34,35)(29,33,36,31);(21,25,26,34)(22,29,30,36)(23,31,28,33)(24,32,27,35)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gh,d^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=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, 597486);