#9050416

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

4316 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8957612);