#1157949

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1502 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=fgi,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1065145);