#2016751

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1779 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,16,15)(17,18)(19,20)(21,23)(22,24)(26,30)(27,33)(28,35)(38,40);(1,2,4,6)(3,5,8,7)(9,10)(11,15)(12,14)(13,16)(17,19,21,24)(18,22,23,20)(26,30)(27,33)(32,37)(34,39);(3,8)(5,7)(9,10,12,14)(11,13,16,15)(17,20,21,22)(18,19,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=hi,c^2=h,d^2=fgh,e^2=g,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, 1923947);