#4680920

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1390 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)(9,10,13,14)(11,12,15,16)(17,18,20,22)(19,23,24,21)(26,27)(28,35)(29,37)(38,39);(1,2)(3,5)(4,6)(7,8)(10,14)(11,15)(25,26,30,27)(28,33,35,31)(29,32,37,34)(36,39,40,38);(3,8)(5,7)(9,11)(10,12)(13,15)(14,16)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,2)(3,5)(4,6)(7,8)(17,19,20,24)(18,21,22,23)(25,28,30,35)(26,31,27,33)(29,36,37,40)(32,38,34,39);(1,3,4,8)(2,5,6,7)(9,12)(10,15)(11,14)(13,16)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

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, 4588116);