#930756

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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

1698 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)(5,8)(9,10,13,15)(11,16,14,12)(17,18)(19,23)(20,24)(21,22)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(11,14)(12,16)(17,19,22,20)(18,23,21,24)(26,30)(31,33)(32,36)(38,39);(1,2,4,6)(3,5,7,8)(9,11,13,14)(10,12,15,16)(17,20,22,19)(18,23,21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,21,22,18)(19,23,20,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=fi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 837952);