#5119752

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1863 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=ghi,c^2=fgh,d^2=gi,e^2=fgh,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, 5026948);