#6131785

order	512
nilpotency class	2
isoclinism	isoclinic class of #6131635
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

1921 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=fg,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg

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, 6038981);