#714306

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×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

1742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,8)(9,10,13,12)(11,14,15,16)(20,24)(22,23)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(11,15)(14,16)(17,18,21,19)(20,22,24,23)(26,31)(27,33)(29,37)(36,39);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,16)(13,15)(17,19,21,18)(20,23,24,22)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,20)(18,22)(19,23)(21,24)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(25,29,30,37)(26,34,31,38)(27,35,33,40)(28,36,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fgi,d^2=gh,e^2=f,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, 621502);