Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1437 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
979 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 979 conjugacy classes of subgroups — too many to draw. Only its 182 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:
(1,2,4,7,5,8,3,6)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,23)(22,24);(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(2,6,8,7)(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239444, #72265657, #72265662, … |
| maximal normal subgroup of | #72239444, #72265657, #72265662, … |
| maximal quotient of | #72257737, #72257745 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17470);