#6135754

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1911 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)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,24,22)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(2,5)(6,8)(11,15)(13,16)(17,18,21,23)(19,20,24,22)(26,30)(28,36)(31,33)(34,37);(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,23,20)(25,27,29,35)(26,31,30,33)(28,37,36,34)(32,40,38,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,28,29,36)(26,32,30,38)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6042950);