Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
976 conjugacy classes of subgroups; only the 234 normal subgroups are shown.
This group has 976 conjugacy classes of subgroups — too many to draw. Only its 234 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:
(2,4)(5,6)(7,11,12,9)(8,10)(13,14,16,20,18,22,17,21)(15,23,24,27,25,28,26,19);(5,7)(6,9)(8,12)(10,11)(13,15,17,24,18,25,16,26)(14,19,21,23,22,27,20,28);(1,2,3,4)(6,10)(9,11)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=eh,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,d^b=di,f^b=fhi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238897, #72238899, #72238900, … |
| maximal normal subgroup of | #72238897, #72238899, #72238900, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 16165);