#772766

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

1862 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=f,d^2=h,e^2=hi,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, 679962);