Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11994 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1558 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8040 |
| prime radical | C24 | #101 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 144 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,4,9,6,8,18,11)(3,12,14,7,16,21,13,23)(5,19,17,10,20,25,28,24)(15,22,29,31,30,32,27,26);(1,3,5,15,18,14,28,29,6,16,20,30,4,13,17,27)(2,7,10,22,9,12,25,26,8,23,24,32,11,21,19,31);(5,20)(10,24)(15,30)(17,28)(19,25)(22,32)(26,31)(27,29)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=gi,e^2=i,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,f^c=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72265710, #72265719 |
| maximal normal subgroup of | #72265710, #72265719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34942);