#9064915

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

4373 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)(11,20)(12,22)(13,19)(14,16)(15,17)(18,23)(21,24)(25,26,30,31)(27,32,36,39)(28,33,37,35)(29,38,40,34);(1,2)(3,4)(5,6)(7,8)(9,11,15,14)(10,16,17,20)(12,21,23,19)(13,18,24,22)(25,26,30,31)(27,32,36,39)(28,35,37,33)(29,34,40,38);(2,6)(3,7)(10,17)(13,24)(16,20)(18,22)(26,31)(28,37)(32,39)(34,38);(3,7)(4,8)(9,12)(10,18)(11,21)(13,20)(14,19)(15,23)(16,24)(17,22)(25,27,30,36)(26,32,31,39)(28,38,37,34)(29,33,40,35);(1,3)(2,4)(5,7)(6,8)(9,13,15,24)(10,19,17,21)(11,22,14,18)(12,16,23,20)(25,28,30,37)(26,33,31,35)(27,34,36,38)(29,39,40,32);(1,4)(2,3)(5,8)(6,7)(9,14,15,11)(10,20,17,16)(12,21,23,19)(13,18,24,22)(25,29)(26,34)(27,35)(28,39)(30,40)(31,38)(32,37)(33,36)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=gh,e^2=h,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, 8972111);