#9159967

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

4621 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=hi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9067163);