Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #124650 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1527 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #376 | C23 |
| socle | C2 | #7609 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 82 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)(3,11,13,27,14,26,19,30)(4,15,17,31)(5,9)(6,10)(7,12,21,28,23,16,22,29)(8,18,24,32)(20,25);(1,3,6,14)(2,7)(4,16,17,12)(5,19,20,13)(8,11)(9,21)(10,23)(15,30)(18,29,32,28)(22,25)(24,26)(27,31);(4,17)(8,24)(11,26)(12,16)(15,31)(18,32)(27,30)(28,29)Polycyclic presentation:
2^9|a^2=e,b^2=g,d^2=gi,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=df,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,d^c=di,e^d=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339723, #72339742, #72339743 |
| maximal normal subgroup of | #72339723, #72339742, #72339743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31879);