#1775050

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

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, 1682246);