#10088566

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 9995762);