#4779033

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

1376 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)(3,5,7,8)(9,10)(11,16)(12,13)(14,15)(18,21)(19,23)(25,26)(27,34)(28,29)(30,31)(32,37)(33,39)(35,36)(38,40);(9,11,14,13)(10,12,15,16)(19,23)(22,24)(26,31)(28,33)(32,34)(35,40);(10,15)(11,13)(17,18,20,21)(19,22,23,24)(25,27,30,37)(26,32,31,34)(28,35,33,40)(29,38,39,36);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,28,30,33)(26,29,31,39)(27,35,37,40)(32,38,34,36);(1,3,4,7)(2,5,6,8)(9,13,14,11)(10,12,15,16)(25,29)(26,33)(27,36)(28,31)(30,39)(32,35)(34,40)(37,38)

Polycyclic presentation:

2^9|a^2=fghi,b^2=h,c^2=gh,d^2=fh,e^2=fgi,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, 4686229);