Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
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×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4778 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)(3,4)(5,6,7,8)(9,10,15,19)(11,21,22,23)(12,20,14,17)(13,18,16,24);(9,11)(10,16)(12,18)(13,19)(14,24)(15,22)(17,21)(20,23);(2,4)(9,12,15,14)(10,17,19,20)(11,18,22,24)(13,23,16,21);(9,13)(10,18)(11,20)(12,23)(14,21)(15,16)(17,22)(19,24);(10,19)(11,22)(17,20)(18,24)Polycyclic presentation:
2^9|a^2=h,c^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,e^b=ei,f^b=fi,f^d=fi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72284104, #72284404, #72315582, … |
| maximal normal subgroup of | #72284104, #72284404, #72315582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286348);