#5956553

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1895 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)(9,10,12,13)(11,15,16,14)(19,23)(22,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,38)(37,40);(2,5)(6,8)(18,21)(22,24)(25,27,30,28)(26,31,33,34)(29,35,37,36)(32,38,40,39);(10,13)(14,15)(17,18)(19,22)(20,21)(23,24)(25,28,30,27)(26,31,33,34)(29,35,37,36)(32,39,40,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(25,29,30,37)(26,32,33,40)(27,35,28,36)(31,38,34,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)(25,26)(27,34)(28,31)(29,32)(30,33)(35,39)(36,38)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=f,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, 5863749);