#6034423

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1680 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)(3,8)(4,7)(5,6)(9,10,13,12)(11,15,16,14)(17,18)(19,20)(21,22)(23,24)(25,26,31,34)(27,32,28,29)(30,38,39,33)(35,37,36,40);(2,5)(7,8)(10,12)(11,16)(18,22)(19,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,40,38,37);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,19,21,24)(18,23,22,20)(25,28,31,27)(26,32,34,29)(30,36,39,35)(33,40,38,37);(1,4,6,3)(2,8,5,7)(9,12,13,10)(11,15,16,14)(17,20,21,23)(18,24,22,19)(25,29,31,32)(26,28,34,27)(30,37,39,40)(33,36,38,35);(1,5,6,2)(3,8,4,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,39)(32,40)(34,38)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=i,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

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, 5941619);