Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129554 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#102 |
| 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 ◁ C43 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
392 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 392 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)(5,6,10,11,9,8,7,12)(13,14,15,22,16,20,23,18)(17,26,24,21,25,19,27,28);(1,2,3,4)(5,7,9,10)(6,11,8,12)(13,15,16,23)(14,18,20,22)(17,24,25,27)(19,21,26,28);(5,8)(6,9)(7,11)(10,12)(14,19)(15,23)(17,27)(18,28)(20,26)(21,22)(24,25)Polycyclic presentation:
2^9|a^2=f,b^2=d,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ch,e^c=egi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318463, #72318464, #72318476, … |
| maximal normal subgroup of | #72318463, #72318464, #72318476, … |
| maximal quotient of | #72322472, #72322542 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41603);