Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #427 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #7898 |
| prime radical | C2×#41 | C22⋊C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #405 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
399 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 399 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:
(1,2,4,8,6,11,20,24,7,12,21,25,5,22,19,10)(3,13,15,27,17,29,32,23,18,31,14,30,16,9,26,28);(2,8,11,24,12,25,22,10)(9,23,13,30,29,28,31,27);(1,3,5,16,7,18,6,17)(2,9,11,13,12,29,22,31)(4,14,19,32,21,15,20,26)(8,23,24,30,25,28,10,27)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=gi,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fgi,g^a=gi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320434, #72320710, #72320735 |
| maximal normal subgroup of | #72320434, #72320710, #72320735 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34068);