#637608

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1958 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)(13,14,17,16)(15,19,20,18)(21,22)(23,29)(24,32)(25,28)(26,31)(27,33)(30,34)(35,36);(5,6,8,10)(7,9,11,12)(13,14,17,16)(15,19,20,18)(21,22,25,28)(23,26,31,29)(24,27,33,32)(30,34,36,35);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,16,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,33)(34,36);(1,2)(3,4)(13,14,17,16)(15,18,20,19)(21,23)(22,26)(24,30)(25,31)(27,34)(28,29)(32,35)(33,36);(5,7,8,11)(6,9,10,12)(13,16,17,14)(15,19,20,18)(21,24)(22,27)(23,30)(25,33)(26,34)(28,32)(29,35)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,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, 544804);