Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139602 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1889 |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | #3492 | C22 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,11,12,15,14,16,13,7)(17,18)(19,25,26,32)(20,28,30,23)(21,24)(22,27)(29,31);(1,3)(2,7,10,15)(4,13)(5,12)(6,14)(8,16,9,11)(23,28)(25,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(17,19,21,26)(18,22,24,31)(20,29,30,27)(23,32,28,25)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=gi,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,h^a=hi,c^b=cg,e^b=ehi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46853);