Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133406 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×D4 | C22×C4 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | #48143 | C2 |
| LCS | G ▷ C22×D4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
870 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 870 conjugacy classes of subgroups — too many to draw. Only its 78 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:
(5,8)(9,10)(11,17)(12,20,22,14)(13,16)(15,18)(19,23,21,24);(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(15,24)(16,20)(17,23)(19,21);(1,3)(2,5)(4,7)(6,8)(9,11,13,18)(10,15)(12,21,22,19)(14,23)(16,17)(20,24)Polycyclic presentation:
2^9|a^2=f,c^2=gi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,c^b=ci,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72232749, #72235232, #72249363, … |
| maximal normal subgroup of | #72232749, #72235232, #72249363, … |
| maximal quotient of | #72232748, #72233622, #72233649, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40711);