#9023597

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

4466 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8930793);