#6051895

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1566 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fghi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5959091);