#9105368

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

4482 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9012564);