#688950

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fh,d^2=fg,e^2=g,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, 596146);