#4463991

order	512
nilpotency class	2
isoclinism	isoclinic class of #4462490
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1382 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)(10,13)(14,16)(19,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,38)(34,35)(36,37)(39,40);(1,3)(2,5)(4,8)(6,7)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(26,28)(31,35)(32,36)(38,40);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(25,27,30,34)(26,31,28,35)(29,33,37,39)(32,38,36,40);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,14,13,16)(17,19)(18,21)(20,23)(22,24)(25,28,30,26)(27,31,34,35)(29,36,37,32)(33,38,39,40);(1,2)(3,5)(4,6)(7,8)(25,29,30,37)(26,32,28,36)(27,33,34,39)(31,38,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4371187);