#764761

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

1860 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)(8,12)(13,14,16,18)(15,17,19,20)(21,22)(23,29)(24,26)(25,32)(27,31)(28,34)(30,35)(33,36);(5,6,10,9)(7,8,11,12)(13,14,16,18)(15,17,19,20)(22,24)(27,29)(28,32)(35,36);(5,7)(6,8)(9,12)(10,11)(13,14)(15,17)(16,18)(19,20)(21,23,26,31)(22,27,24,29)(25,33,34,30)(28,36,32,35);(1,2)(3,4)(5,8,10,12)(6,11,9,7)(21,24,26,22)(23,29,31,27)(25,32,34,28)(30,36,33,35);(5,9,10,6)(7,12,11,8)(13,15,16,19)(14,17,18,20)(21,25,26,34)(22,28,24,32)(23,30,31,33)(27,35,29,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=f,d^2=gh,e^2=g,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, 671957);