#9065937

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

4392 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973133);