#5206749

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1493 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)(3,6)(4,8)(5,7)(9,10,13,16)(11,14,12,15)(19,23)(21,24)(25,26,29,30)(27,31,33,37)(28,35,36,32)(34,40,39,38);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,16,15)(17,18,20,22)(19,21,23,24)(26,30)(27,33)(28,36)(38,40);(2,7)(3,8)(9,11,13,12)(10,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,37)(34,40)(38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,15)(13,16)(25,27,29,33)(26,31,30,37)(28,34,36,39)(32,38,35,40);(1,4)(2,3)(5,6)(7,8)(9,12,13,11)(10,15,16,14)(17,19,20,23)(18,21,22,24)(25,28)(26,32)(27,34)(29,36)(30,35)(31,38)(33,39)(37,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=fgi,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5113945);