Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10518 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #394 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10,13,16,14,17,12,20)(11,15,18,24,19,23,21,22);(11,18,19,21)(15,22,23,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,18)(14,19)(16,24)(17,23)(20,22)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^c=dh,e^c=ei,f^c=fh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239518, #72239519, #72257466 |
| maximal normal subgroup of | #72239518, #72239519, #72257466 |
| maximal quotient of | #72239500, #72257605 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23596);