#9028696

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

4580 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)(11,18)(12,15)(13,16)(14,17)(19,21)(20,23)(22,24)(25,26)(27,33)(28,36)(29,32)(30,35)(31,37)(34,38)(39,40);(1,2,5,4)(3,7,8,6)(9,11)(10,12)(13,19)(14,15)(16,20)(17,18)(21,24)(22,23)(25,26,29,32)(27,30,35,33)(28,31,37,36)(34,38,40,39);(1,3,5,8)(2,6,4,7)(9,12,14,18)(10,15,17,11)(13,21,22,20)(16,19,24,23)(25,26,29,32)(27,33,35,30)(28,36,37,31)(34,38,40,39);(3,8)(6,7)(9,12)(10,11)(13,21)(14,18)(15,17)(16,23)(19,24)(20,22)(25,27,29,35)(26,30,32,33)(28,34,37,40)(31,38,36,39);(1,3)(2,7)(4,6)(5,8)(9,13,14,22)(10,16,17,24)(11,19,15,23)(12,20,18,21)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,4,5,2)(3,6,8,7)(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,28)(26,31)(27,34)(29,37)(30,38)(32,36)(33,39)(35,40)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=h,e^2=gh,f^2=i,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, 8935892);