Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395166 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#493 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(10,12)(11,19)(13,14)(16,23)(17,18)(20,24);(1,2,4,3)(5,6)(7,8)(9,10,14,17,15,18,13,12)(11,16,22,23,19,24,21,20);(1,3,4,2)(6,8)(9,11,13,21,15,19,14,22)(10,16,12,20,18,24,17,23);(9,12,13,18,15,17,14,10)(11,20,21,24,19,23,22,16)Polycyclic presentation:
2^9|b^2=ei,c^2=e,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,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226904, #72226919, #72226922, … |
| maximal normal subgroup of | #72226904, #72226919, #72226922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319347);