Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141213 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1972 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | #33748 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(10,16,20,19)(11,22,21,23)(12,17)(13,18)(14,24);(1,2,3,7,4,6,8,5)(9,10,17,23,24,11,12,19,15,20,18,22,14,21,13,16);(9,11,13,22,14,10,18,19,15,21,12,23,24,20,17,16)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=g,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227102, #72227130, #72227177, … |
| maximal normal subgroup of | #72227102, #72227130, #72227177, … |
| maximal quotient of | #72227094 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48409);