#5719200

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=hi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5626396);