Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35565 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1867 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1867 conjugacy classes of subgroups — too many to draw. Only its 210 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,8)(9,10)(11,17)(12,20)(13,16)(14,18)(15,21)(19,24)(22,23);(1,2,4,5)(3,6,7,8)(12,21)(14,23)(15,20)(18,22);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,22)(13,19)(15,24)(16,23)(17,20)(18,21);(2,5)(6,8)(12,21)(15,20)(17,24)(18,22)Polycyclic presentation:
2^9|b^2=e,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72256724, #72256727, #72256728 |
| maximal normal subgroup of | #72256724, #72256727, #72256728 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325322);