#1324656

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1614 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=fgi,d^2=fhi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

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, 1231852);