#9161752

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

4329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gh,e^2=ghi,f^2=gh,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, 9068948);