#1369341

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
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

1668 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=f,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1276537);