Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1074 conjugacy classes of subgroups; only the 378 normal subgroups are shown.
This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 378 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) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(21,22)(23,31)(24,27)(25,35)(26,30)(28,33)(29,32)(34,36);(6,9)(10,12)(13,14)(15,18)(16,17)(19,20)(21,23,32,34,26,33,25,24)(22,27,35,28,30,36,29,31);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(21,24,25,33,26,34,32,23)(22,28,29,27,30,31,35,36);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=e,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266151);