Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #100203 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
2248 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 2248 conjugacy classes of subgroups — too many to draw. Only its 154 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)(7,11,12,9)(8,10)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(5,7)(6,9)(8,12)(10,11)(18,24)(25,28);(2,4)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28)Polycyclic presentation:
2^9|a^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,g^b=gh,d^c=di62 conjugacy classes — view the full character table.
| maximal subgroup of | #72235200, #72235224, #72238444, … |
| maximal normal subgroup of | #72235200, #72235224, #72238444, … |
| maximal quotient of | #72235926 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22337);