#9236461

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

4318 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)(3,4)(5,7)(6,8)(9,10,15,20)(11,16,19,14)(12,18,13,17)(21,23,22,24)(27,33)(30,34)(35,39)(37,40);(1,3,6,5)(2,7,8,4)(9,11,15,19)(10,16,20,14)(12,22,13,21)(17,23,18,24)(25,26,31,28)(27,30,33,34)(29,36,38,32)(35,40,39,37);(9,12,15,13)(10,17,20,18)(11,21,19,22)(14,23,16,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,4)(2,3)(5,8)(6,7)(9,13,15,12)(10,18,20,17)(11,21,19,22)(14,23,16,24)(25,28,31,26)(27,34,33,30)(29,36,38,32)(35,40,39,37);(1,5,6,3)(2,7,8,4)(9,11)(10,16)(12,21)(13,22)(14,20)(15,19)(17,24)(18,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40);(1,2)(3,7)(4,5)(6,8)(9,14,15,16)(10,19,20,11)(12,23,13,24)(17,22,18,21)(25,30,31,34)(26,33,28,27)(29,37,38,40)(32,39,36,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143657);