Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 512 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,7,8,10,12,11)(14,18)(15,22)(16,23)(19,25)(20,24)(21,28)(26,27);(2,4)(5,7,9,10,8,11,12,6)(14,19)(18,25)(21,26)(27,28);(13,14,20,28,16,21,15,18,17,19,22,27,23,26,24,25)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=egi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286684, #72286742, #72320179, … |
| maximal normal subgroup of | #72286684, #72286742, #72320179, … |
| maximal quotient of | #72286771, #72320488 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47540);