#9064917

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

4365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(9,10)(11,19)(12,17)(13,20)(14,15)(16,22)(18,24)(21,23)(25,26,30,31)(27,32,36,39)(28,33,37,35)(29,38,40,34);(1,2)(3,4)(5,6)(7,8)(10,15)(12,23)(16,19)(18,20)(25,26,30,31)(27,32,36,39)(28,35,37,33)(29,34,40,38);(2,6)(3,7)(10,15)(12,23)(16,19)(18,20)(26,31)(28,37)(32,39)(34,38);(3,7)(4,8)(9,11)(10,16)(12,18)(13,17)(14,22)(15,19)(20,23)(21,24)(25,27,30,36)(26,32,31,39)(28,38,37,34)(29,33,40,35);(1,3)(2,4)(5,7)(6,8)(9,12,14,23)(10,17,15,21)(11,20,22,18)(13,16,24,19)(25,28,30,37)(26,33,31,35)(27,34,36,38)(29,39,40,32);(1,4)(2,3)(5,8)(6,7)(9,13,14,24)(10,18,15,20)(11,21,22,17)(12,16,23,19)(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=gh,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, 8972113);