Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135932 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C22×D8 | C2×C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 90 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:
(1,2)(3,4)(5,6,9,11,8,7,10,12)(14,20,23,22)(15,24,26,25)(16,27)(17,21)(18,28);(5,7,9,12,8,6,10,11)(13,14,21,25,28,26,17,20,19,23,27,24,18,15,16,22);(2,4)(6,7)(11,12)(13,15,16,25,18,23,27,22,19,26,17,24,28,14,21,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=d,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cgh,d^b=di,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318613, #72318619, #72318646, … |
| maximal normal subgroup of | #72318613, #72318619, #72318646, … |
| maximal quotient of | #72320870, #72320917, #72322414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47406);