#9236497

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

4841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=i,e^2=g,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, 9143693);