Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100139 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1884 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 1884 conjugacy classes of subgroups — too many to draw. Only its 162 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) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14)(15,19)(16,18)(17,21)(20,24)(22,27)(23,28)(25,26);(5,7)(6,9)(8,12)(10,11)(14,18)(16,24)(19,27)(20,21)(22,25)(26,28);(2,4)(13,15,17,23)(14,19,21,28)(16,25,24,22)(18,26,20,27)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,g^b=ghi,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72314409 |
| maximal normal subgroup of | #72314409 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19089);