Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #46318 | C2 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
720 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 720 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)(12,16,20,15)(13,21)(14,18)(19,23,22,24);(1,2)(3,5)(4,6)(7,8)(10,15)(11,13)(12,20)(16,18)(17,23)(21,24);(1,3)(2,5)(4,7)(6,8)(9,11,14,13)(12,19,20,22)(15,16)(23,24)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72261096, #72261111, #72261118 |
| maximal normal subgroup of | #72261096, #72261111, #72261118 |
| maximal quotient of | #72261639 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40657);