#9506936

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

4640 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9414132);