#707446

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1798 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fhi,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, 614642);