Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#399 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1086 conjugacy classes of subgroups; only the 298 normal subgroups are shown.
This group has 1086 conjugacy classes of subgroups — too many to draw. Only its 298 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 — 4 generators on 36 points:
(1,2,4,3)(5,6,11,16,10,15,9,14)(7,17,20,12,21,28,19,27)(8,22,25,30,23,13,24,31)(18,32,35,26,33,36,34,29);(5,7,11,19,10,21,9,20)(6,12,16,17,15,27,14,28)(8,18,25,34,23,33,24,35)(13,29,31,36,22,26,30,32);(5,8)(6,13)(7,18)(9,24)(10,23)(11,25)(12,26)(14,30)(15,22)(16,31)(17,32)(19,34)(20,35)(21,33)(27,29)(28,36);(1,3,4,2)(8,23)(13,22)(18,33)(24,25)(26,29)(30,31)(32,36)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266484);