#4928734

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1638 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,16)(14,15)(19,23)(22,24)(25,26,29,30)(27,33,34,31)(28,36,38,32)(35,39,37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18,21,20)(19,22,23,24)(26,30)(27,34)(32,36)(35,37);(9,12)(10,15)(11,16)(13,14)(17,18,21,20)(19,22,23,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,29,34)(26,31,30,33)(28,37,38,35)(32,40,36,39);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20,21,18)(19,24,23,22)(25,28,29,38)(26,32,30,36)(27,35,34,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=fghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=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, 4835930);