Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#120 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1157 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 1157 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4,7,5)(6,8)(9,10,13,17,14,16,12,22)(11,18,21,23,20,24,19,15);(1,3,4,8)(2,6,7,5)(9,11)(10,15,16,23)(12,19)(13,21)(14,20)(17,18,22,24);(1,4)(2,7)(3,8)(5,6)(10,16)(12,13)(15,23)(19,21)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,f^c=fgh,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72232567, #72232570, #72232574, … |
| maximal normal subgroup of | #72232567, #72232570, #72232574, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46680);