Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7349 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1317 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | #379 | C23 |
| socle | C2 | C2×#1317 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #379 ▷ 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:
(1,2,5,10)(3,7,13,19,12,20,16,28)(4,8,14,22)(6,9,17,23)(11,18,25,29,24,30,26,32)(15,21,27,31);(1,3)(2,7,9,20)(4,11)(5,16)(6,12)(8,18,21,30)(10,19,23,28)(13,17)(14,26)(15,24)(22,29,31,32)(25,27);(1,4,6,15)(2,8,9,21)(3,11,12,24)(5,14,17,27)(7,18,20,30)(10,22,23,31)(13,25,16,26)(19,29,28,32)Polycyclic presentation:
2^9|a^2=e,b^2=h,c^2=i,e^2=g,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=di,e^b=ef,g^b=gi,e^d=eh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339710, #72339727, #72339729 |
| maximal normal subgroup of | #72339710, #72339727, #72339729 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56748);