#9206623

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

4508 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,d^2=h,e^2=ghi,f^2=gh,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, 9113819);