#6269792

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	C₂²×#121	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=fhi,d^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, 6176988);