#4277868

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1438 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)(11,15)(12,16)(13,14)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,40)(38,39);(9,11,14,12)(10,15,13,16)(18,21)(22,24)(26,27)(29,37)(31,33)(35,38);(9,10,14,13)(11,15,12,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,15,13,16)(25,28,30,36)(26,31,27,33)(29,38,37,35)(32,40,34,39);(1,3,4,7)(2,5,6,8)(9,13,14,10)(11,15,12,16)(17,19,20,23)(18,22,21,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,39)(33,40)(36,38)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fg,c^2=g,d^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4185064);