#5958485

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1539 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fg,d^2=fg,e^2=g,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, 5865681);