#6269740

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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

1750 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,20,24,23)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(3,8)(4,7)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,24)(21,22)(27,34)(29,37)(31,33)(32,36);(2,6)(3,8)(10,14)(13,15)(18,22)(20,23)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,16)(13,14)(17,19)(18,20)(21,24)(22,23)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,4,5,7)(2,8,6,3)(9,10)(11,15)(12,14)(13,16)(17,20)(18,19)(21,23)(22,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=fgi,d^2=fg,e^2=fg,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, 6176936);