Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1316 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | #381 | C23 |
| socle | C2 | C2×#1316 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
1086 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 80 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:
(1,2,6,10)(3,12,14,29,15,22,30,32)(4,9,16,25)(5,17,19,27)(7,11,20,28)(8,21,23,18,24,31,26,13);(1,3)(2,8)(4,13)(5,18)(6,14)(7,15)(9,22)(10,26)(11,24)(12,17)(16,21)(19,31)(20,30)(23,28)(25,29)(27,32);(1,4,7,5)(2,9,11,17)(3,13,15,18)(6,16,20,19)(8,22,24,12)(10,25,28,27)(14,21,30,31)(23,32,26,29)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=i,e^2=g,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=efi,g^b=gi,e^d=eh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339710, #72339725, #72339726 |
| maximal normal subgroup of | #72339710, #72339725, #72339726 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56726);