Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94311 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C2×C4⋊C4 | C42 |
| Frattini, cosocle | C2×#384 | C22 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#384 ◁ G | |
| LECS | G ▷ C2×#384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17,19,24)(12,15)(13,22)(14,18,23,20)(16,21);(2,5,7,6)(9,11)(10,14,16,17,15,23,22,24)(12,19)(13,18)(20,21)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,e^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=efg,g^a=gi,h^a=hi,c^b=cg,d^b=di,g^b=gh,e^c=ehi,g^c=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1524);