#5989640

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2003 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=f,d^2=fg,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, 5896836);