#9206617

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

4336 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=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, 9113813);