Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #102905 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C22×C42 | C23 |
| prime radical | #17068 | C2 |
| LCS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
991 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 991 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,17)(16,18)(19,20)(22,25)(26,27);(1,2)(3,4)(6,9)(10,11)(13,15)(14,17)(16,19)(18,20)(21,22)(23,25)(24,26)(27,28);(1,3,4,2)(6,10)(8,12)(9,11)(17,20)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|c^2=dgi,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72217742 |
| maximal normal subgroup of | #72217742 |
| maximal quotient of | #72215344, #72215350, #72216328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12019);