Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#535 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(6,8)(9,10,14,17,15,11,13,18)(12,16,22,23,20,19,21,24);(1,2,3,4)(9,11,15,10)(12,19,20,16)(13,18,14,17)(21,24,22,23);(5,6,7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24);(12,20)(16,19)(21,22)(23,24);(9,13,15,14)(10,17,11,18)(12,21,20,22)(16,23,19,24)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^b=fi,h^b=hi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72312721, #72312751, #72313019, … |
| maximal normal subgroup of | #72312721, #72312751, #72313019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279248);