Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4922 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)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(2,4)(9,11)(10,15)(12,20)(13,21)(14,22)(16,19)(17,23)(18,24);(5,6,7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,19,24,23)Polycyclic presentation:
2^9|c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257585, #72264363, #72264439, … |
| maximal normal subgroup of | #72257585, #72264363, #72264439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349824);