#764621

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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

1902 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)(7,10,11,12)(13,14)(15,16)(17,19)(18,20)(21,22,26,23)(24,29,31,27)(25,30,33,28)(32,34,36,35);(7,11)(10,12)(13,15,17,20)(14,16,19,18)(22,23)(24,31)(28,30)(32,36);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,23,26,22)(24,29,31,27)(25,28,33,30)(32,34,36,35);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,17,18)(14,15,19,20)(21,24)(22,27)(23,29)(25,32)(26,31)(28,34)(30,35)(33,36);(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)

Polycyclic presentation:

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

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, 671817);