#4780119

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1652 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=fg,e^2=fghi,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, 4687315);