#4928749

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1603 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ghi,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, 4835945);