#2618881

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1719 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2526077);