Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1439 |
| prime radical | C23×D4 | Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
798 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 798 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,5)(4,7)(6,8)(9,10,13,16,14,17,12,22)(11,18,21,23,20,24,19,15);(3,8)(5,6)(9,11,12,19,14,20,13,21)(10,15,16,18,17,23,22,24);(2,5)(3,8)(6,7)(15,23)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308318, #72308364, #72308380 |
| maximal normal subgroup of | #72308318, #72308364, #72308380 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23798);