#9064872

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

4649 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)(11,16)(12,18)(13,14)(15,17)(19,24)(20,23)(21,22)(25,26)(27,33)(28,32)(29,34)(30,35)(31,36)(37,38)(39,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,18)(10,16,17,12)(13,22,19,20)(14,21,24,23)(25,27,31,28)(26,32,36,33)(29,37,35,39)(30,38,34,40);(2,6)(3,7)(10,17)(11,18)(13,19)(21,23)(25,27,31,28)(26,33,36,32)(29,39,35,37)(30,38,34,40);(3,7)(4,8)(9,12,15,16)(10,18,17,11)(13,23,19,21)(14,20,24,22)(25,28,31,27)(26,33,36,32)(29,37,35,39)(30,38,34,40);(1,3)(2,4)(5,7)(6,8)(9,13,15,19)(10,14,17,24)(11,20,18,22)(12,21,16,23)(25,29,31,35)(26,34,36,30)(27,37,28,39)(32,40,33,38);(1,4)(2,3)(5,8)(6,7)(9,14)(10,19)(11,21)(12,22)(13,17)(15,24)(16,20)(18,23)(25,30,31,34)(26,35,36,29)(27,38,28,40)(32,39,33,37)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gh,e^2=gh,f^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, 8972068);