#9161715

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

4657 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gh,e^2=hi,f^2=gh,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, 9068911);