Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111399 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#469 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1316 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1316 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10)(11,18,21,23)(12,16)(13,22)(14,17)(15,19,24,20);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,19)(14,21)(16,23)(17,24)(18,22);(1,3)(2,5)(4,8)(6,7)(11,19)(12,13)(15,23)(16,22)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=g,d^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=di,f^c=fi,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72216083, #72216126, #72216128, … |
| maximal normal subgroup of | #72216083, #72216126, #72216128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18595);