Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4⋊Q8 | C23 |
| socle | C22 | C2×#353 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C2×C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
658 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 658 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,9,14,10,15,24,31)(7,16,18,30,19,25,34,35)(8,12,20,28,21,13,23,29)(11,22,26,32,27,33,36,17);(1,2,3,4)(5,7,8,17,10,19,21,32)(6,11,13,16,15,27,12,25)(9,22,23,18,24,33,20,34)(14,30,28,26,31,35,29,36);(5,8,10,21)(6,12,15,13)(7,17,19,32)(9,20,24,23)(11,25,27,16)(14,28,31,29)(18,22,34,33)(26,35,36,30)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bd,d^a=dg,f^a=fhi,g^a=gh,h^a=hi,e^b=egh,e^d=ehi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59034);