Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #98566 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | #59126 | C2 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
1526 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1526 conjugacy classes of subgroups — too many to draw. Only its 130 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:
(1,2,5,6,4,3,7,8)(9,10,12,14,13,16,11,15)(17,18)(19,23,24,21)(20,22);(5,7)(6,8)(10,14)(11,12)(15,16);(1,3)(2,4)(5,6)(7,8)(17,19)(18,21)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg,f^c=fgi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72219584, #72219625, #72224609 |
| maximal normal subgroup of | #72219584, #72219625, #72224609 |
| maximal quotient of | #72219579, #72219582, #72219623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5814);