#6135737

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
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

1772 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)(11,15)(12,13)(14,16)(17,18,20,22)(19,21,24,23)(25,26)(27,32)(28,34)(29,33)(30,31)(35,40)(36,37)(38,39);(2,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,20,24)(18,21,22,23)(26,31)(27,34)(33,39)(35,37);(10,13)(14,15)(18,22)(21,23)(25,27,30,34)(26,32,31,28)(29,37,38,35)(33,36,39,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,22)(21,24)(25,28,30,32)(26,27,31,34)(29,36,38,40)(33,35,39,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,24)(21,22)(25,29,30,38)(26,33,31,39)(27,35,34,37)(28,36,32,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=f,e^2=f,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, 6042933);