Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
2060 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 2060 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(13,14)(15,19)(16,23,24,18)(17,20)(21,27,28,25)(22,26);(14,18)(19,25)(20,23)(26,27);(2,4)(13,15,17,22)(14,19,20,26)(16,21,24,28)(18,25,23,27)Polycyclic presentation:
2^9|a^2=f,c^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,f^b=fg,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284451, #72284996, #72285169, … |
| maximal normal subgroup of | #72284451, #72284996, #72285169, … |
| maximal quotient of | #72293138, #72293156, #72338317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36407);