#1241735

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

1496 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fgh,d^2=g,e^2=fghi,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, 1148931);