#9061580

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

4460 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968776);