Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #130722 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1786 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×D4×D8 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1026 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,15,14,18,11,19,13,17,20,23,21,24,22,16);(2,5)(3,7)(10,15)(11,20)(12,22)(14,21)(16,18)(17,23)(19,24);(2,5)(3,7)Polycyclic presentation:
2^9|a^2=f,d^2=hi,f^2=ghi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,f^b=fgh,h^b=hi,f^c=fgi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72265329, #72308613, #72308616, … |
| maximal normal subgroup of | #72265329, #72308613, #72308616, … |
| maximal quotient of | #72343287, #72343414 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 38240);