#4442764

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1432 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)(3,6,4,7)(11,15)(14,16)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,35)(29,30)(32,37)(33,40)(34,38)(36,39);(1,3,5,4)(2,6,8,7)(10,13)(14,16)(17,18,21,20)(19,22,24,23)(26,30)(31,38)(32,35)(39,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,38)(35,37)(36,40);(1,2)(3,6)(4,7)(5,8)(17,18,21,20)(19,22,24,23)(25,27,29,34)(26,31,30,38)(28,36,37,33)(32,40,35,39);(1,4,5,3)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,18)(19,23,24,22)(25,28,29,37)(26,32,30,35)(27,33,34,36)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=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, 4349960);