#6051900

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

1510 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=ghi,e^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, 5959096);