#757308

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=f,d^2=hi,e^2=fhi,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, 664504);