#714392

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

1582 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=h,e^2=fg,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, 621588);