Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149927 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #25734 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C23×C4 ◁ C2×#535 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
642 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 642 conjugacy classes of subgroups — too many to draw. Only its 68 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:
(1,2)(3,4)(5,6)(7,17,18,31)(8,16,10,13)(9,23,26,30)(11,15)(12,19,28,35)(14,25)(20,34,21,33)(22,27)(24,29,36,32);(5,7,9,24,11,20,26,35)(6,12,27,33,15,29,14,31)(8,18,25,36,10,21,22,19)(13,28,30,17,16,32,23,34);(2,4)(5,8,11,10)(6,13,15,16)(7,18,20,21)(9,25,26,22)(12,28,29,32)(14,23,27,30)(17,31,34,33)(19,24,36,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=i,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dghi,e^b=ei,f^b=fghi,g^b=ghi,h^b=hi,d^c=di,f^c=fi,f^d=fhi,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59957);