#644673

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

1966 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)(13,14,17,15)(16,19,20,18)(21,22,26,30)(23,28,24,27)(25,29,33,36)(31,35,32,34);(5,6,8,10)(7,9,11,12)(21,23,26,24)(22,27,30,28)(25,31,33,32)(29,34,36,35);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,14)(16,19,20,18)(21,24,26,23)(22,28,30,27)(25,31,33,32)(29,34,36,35);(13,16,17,20)(14,18,15,19)(21,25,26,33)(22,29,30,36)(23,31,24,32)(27,34,28,35);(5,7,8,11)(6,9,10,12)(21,22)(23,27)(24,28)(25,29)(26,30)(31,34)(32,35)(33,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=gi,d^2=fi,e^2=h,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, 551869);