Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C4×D8 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
394 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 114 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:
(2,4)(5,6,10,11,9,7,8,12)(14,18)(15,19)(16,22)(20,28)(21,26)(23,27)(24,25);(1,2,3,4)(5,7,10,12,9,6,8,11)(13,14,19,25,16,20,23,26,17,21,27,28,22,24,15,18);(5,8,9,10)(6,11,7,12)(13,14,19,25,16,20,23,26,17,21,27,28,22,24,15,18)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eg,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320185, #72320199, #72320212 |
| maximal normal subgroup of | #72320185, #72320199, #72320212 |
| maximal quotient of | #72320539, #72320634 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47962);