#9017314

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

4612 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,21)(14,16)(15,17)(18,22)(19,23)(20,24)(26,31)(27,34)(33,39)(35,37);(2,6)(4,7)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,23,21,20)(25,26,30,31)(27,32,34,29)(28,33,36,39)(35,40,37,38);(1,2,5,6)(3,7,8,4)(10,16)(12,15)(13,22)(19,24)(27,34)(28,36)(29,32)(33,39);(1,3)(2,7)(4,6)(5,8)(9,12,14,15)(10,17,16,11)(13,20,22,23)(18,24,21,19)(25,27)(26,32)(28,37)(29,31)(30,34)(33,38)(35,36)(39,40);(1,4,5,7)(2,3,6,8)(9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,22)(23,24)(25,28,30,36)(26,33,31,39)(27,35,34,37)(29,38,32,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,19)(12,20)(14,22)(15,23)(16,21)(17,24)(25,29,30,32)(26,27,31,34)(28,38,36,40)(33,35,39,37)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=hi,e^2=ghi,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^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924510);