Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,18)(19,20)(21,24)(22,23);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(18,19)(20,22)(23,24);(9,12,13,11)(10,16,15,14)(17,19)(18,22)(20,24)(21,23);(1,4,5,3)(2,8,6,7)(18,19)(20,22)(23,24)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72305305, #72305306, #72305313, … |
| maximal normal subgroup of | #72305305, #72305306, #72305313, … |
| maximal quotient of | #72306128, #72306148, #72315934 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6497344);