#6253900

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=gh,c^2=fgi,d^2=g,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, 6161096);