Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111825 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #60415 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10,14,17,15,12,13,18)(11,19,21,16,22,20,24,23);(2,6)(3,4)(7,8)(9,11)(10,16)(12,23)(13,24)(14,21)(15,22)(17,20)(18,19);(9,12)(10,15)(11,20)(13,18)(14,17)(16,21)(19,22)(23,24)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,e^c=eh,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280202, #72280211, #72280232 |
| maximal normal subgroup of | #72280202, #72280211, #72280232 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19024);