#9161164

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

4505 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:

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

Polycyclic presentation:

2^9|b^2=gh,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9068360);