Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
601 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 601 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,4,3)(5,6)(7,10)(8,11)(9,12)(13,14)(15,22)(16,18)(17,21)(19,23)(20,25)(24,28)(26,27);(1,3,4,2)(6,10)(7,8)(11,12)(14,18)(15,23)(16,25)(19,28)(20,21)(22,27);(6,11)(7,8)(10,12)(13,15,17,23)(14,19)(16,26,25,24)(18,27)(20,28)(21,22)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,d^2=g,e^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,d^c=dh,e^c=eg,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43381);