#9236450

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

4216 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=gh,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, 9143646);