Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3387 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(9,10)(11,21)(12,16)(13,18)(14,20)(15,19)(17,23)(22,24);(1,2,4,3)(5,6)(7,8)(9,11)(10,16)(12,14)(13,23)(15,22)(17,19)(18,24)(20,21);(1,3,4,2)(6,8)(9,12,14,11)(10,17,20,24)(13,23,15,22)(16,18,21,19);(6,8)(9,13,14,15)(10,18,20,19)(11,22,12,23)(16,17,21,24);(9,13,14,15)(10,19,20,18)(11,22,12,23)(16,24,21,17)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72316106, #72316126, #72316163 |
| maximal normal subgroup of | #72316106, #72316126, #72316163 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345998);