Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #96971 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1957 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 1957 conjugacy classes of subgroups — too many to draw. Only its 170 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:
(5,6)(7,11,12,9)(8,10)(14,18,20,23)(19,27,28,26);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,23)(17,20)(18,25)(21,26)(22,28)(24,27);(2,4)(13,15,17,22)(14,19,20,28)(16,24,25,21)(18,26,23,27)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72266612 |
| maximal normal subgroup of | #72266612 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 20235);