Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19745 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
934 conjugacy classes of subgroups; only the 238 normal subgroups are shown.
This group has 934 conjugacy classes of subgroups — too many to draw. Only its 238 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 — 4 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(10,12)(13,14)(15,21)(16,24)(17,19)(18,23)(20,25)(22,27)(26,28);(13,15,16,23,17,22,25,28)(14,18,20,21,19,26,24,27);(5,6,7,9,8,10,11,12)(14,19)(18,26)(20,24)(21,27);(1,3,4,2)(6,10)(9,12)(15,22)(18,26)(21,27)(23,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216602, #72224761, #72224769, … |
| maximal normal subgroup of | #72216602, #72224761, #72224769, … |
| maximal quotient of | #72225294, #72225296, #72242460 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120118);