Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C42 |
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 | #15763 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
910 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 910 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,4,6,8)(5,7)(9,10)(11,17)(12,20,22,15)(13,19)(14,16)(18,21,24,23);(1,3,8,7,5,6,4,2)(9,11,17,10,14,19,13,16)(12,18,21,15,22,24,23,20);(1,4,5,8)(2,6,7,3)(9,12)(10,15)(11,18)(13,23)(14,22)(16,20)(17,21)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^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=fhi,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 | #72213599, #72235368, #72235390 |
| maximal normal subgroup of | #72213599, #72235368, #72235390 |
| maximal quotient of | #72267646 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46653);