#9065934

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

4655 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8973130);