#9003153

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

4459 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8910349);