Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2036 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(10,15)(11,18)(12,14)(16,21)(17,23)(19,20);(5,6)(7,8)(9,10)(11,17)(12,21)(13,16)(14,15)(18,22)(19,23)(20,24);(2,4)(6,8)(10,16)(11,19)(15,21)(18,20);(9,11,12,20,13,19,14,18)(10,17,21,24,16,23,15,22)Polycyclic presentation:
2^9|d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72226898, #72226901, #72226912, … |
| maximal normal subgroup of | #72226898, #72226901, #72226912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318842);