#9159937

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

4593 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=ghi,e^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, 9067133);