Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7348 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1317 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | #381 | C23 |
| socle | C2 | C2×#1317 |
| prime radical | C25 | C22⋊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 | |
462 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 462 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)(3,12,15,28,16,29,31,32)(4,9,17,24)(5,18,19,26)(7,11,20,27)(8,21,22,13,23,30,25,14);(1,3)(2,8,11,23)(4,13)(5,14)(6,15)(7,16)(9,12,18,29)(10,25,27,22)(17,30)(19,21)(20,31)(24,32,26,28);(1,4,7,5)(2,9,11,18)(3,13,16,14)(6,17,20,19)(8,12,23,29)(10,24,27,26)(15,30,31,21)(22,28,25,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=i,d^2=i,e^2=g,b^a=bd,d^a=df,f^a=fh,h^a=hi,e^b=efi,g^b=gi,e^d=eh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338317, #72338382 |
| maximal normal subgroup of | #72338317, #72338382 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56746);