#4407863

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
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

1438 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)(11,15)(14,16)(19,20)(23,24)(25,26,29,30)(27,31,34,37)(28,32,36,38)(33,39,35,40);(9,10,13,12)(11,14,15,16)(18,22)(23,24)(26,30)(31,37)(32,38)(39,40);(9,11)(10,14)(12,16)(13,15)(17,18,21,22)(19,23,20,24)(25,27,29,34)(26,31,30,37)(28,35,36,33)(32,40,38,39);(1,2)(3,6)(4,5)(7,8)(17,19,21,20)(18,23,22,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,40)(36,38);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,24,22,23)(25,28,29,36)(26,32,30,38)(27,33,34,35)(31,39,37,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4315059);