#8936732

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

4617 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:

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

Polycyclic presentation:

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

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, 8843928);