Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #108534 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C4⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1770 |
| prime radical | C23 | #436 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M32 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10,13,11,12,16,19,18,14,17,22,20,21,15,23,24);(2,6)(3,4)(7,8)(10,15,17,16)(11,18,20,24)(13,22)(19,23);(1,3,5,4)(2,7,8,6)(9,11,19,17,21,24,13,16,14,20,23,10,12,18,22,15)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=gi,e^2=h,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225552, #72226987, #72227103, … |
| maximal normal subgroup of | #72225552, #72226987, #72227103, … |
| maximal quotient of | #72225535, #72227184 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36354);