#9160748

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

4516 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,18)(13,19)(14,22)(15,17)(20,23)(21,24)(25,26,31,29)(27,36,38,32)(28,33,39,34)(30,35,40,37);(1,2,5,4)(3,6,8,7)(9,11,15,20)(10,16,17,23)(12,21,22,13)(14,19,18,24)(25,27,31,38)(26,32,29,36)(28,37,39,35)(30,33,40,34);(1,3)(2,6)(4,7)(5,8)(10,17)(11,20)(12,22)(19,24)(25,28,31,39)(26,33,29,34)(27,37,38,35)(30,32,40,36);(1,3,5,8)(2,6,4,7)(9,12)(10,18)(11,21)(13,20)(14,17)(15,22)(16,24)(19,23)(25,28)(26,34)(27,37)(29,33)(30,32)(31,39)(35,38)(36,40);(1,3)(2,7)(4,6)(5,8)(9,13,15,21)(10,19,17,24)(11,22,20,12)(14,23,18,16)(25,29,31,26)(27,36,38,32)(28,34,39,33)(30,35,40,37);(1,4,5,2)(3,7,8,6)(9,14,15,18)(10,12,17,22)(11,19,20,24)(13,16,21,23)(25,30)(26,35)(27,33)(28,32)(29,37)(31,40)(34,38)(36,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=hi,e^2=gh,f^2=h,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, 9067944);