Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13603 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#354 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | #532 | C23 |
| socle | C2 | C2×#1393 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
647 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 128 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:
(1,2,4,9,6,10,17,27)(3,12,14,26,13,24,29,32)(5,8,19,25,20,11,18,28)(7,21,23,15,22,16,31,30);(1,3)(2,7,10,22)(4,16,17,21)(5,15)(6,13)(8,12,11,24)(9,26)(14,19,29,18)(20,30)(23,28)(25,31)(27,32);(3,13)(7,22)(12,24)(14,29)(15,30)(16,21)(23,31)(26,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=h,e^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fh,g^a=gh,h^a=hi,c^b=ci,e^b=eghi,e^d=ehi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72339653, #72339655, #72339656, … |
| maximal normal subgroup of | #72339653, #72339655, #72339656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59040);