Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C42 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C23 | #407 |
| LCS | G ▷ C42 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 132 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:
(1,2,5,6)(3,4,7,8)(10,15,16,18)(12,20)(17,24,23,22)(19,21);(1,3,4,6,5,7,8,2)(9,10,12,17,13,18,21,24,14,16,20,23,11,15,19,22);(10,15,16,18)(12,20)(17,24,23,22)(19,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,f^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=cgh,e^b=ei,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72309481, #72309489, #72309501, … |
| maximal normal subgroup of | #72309481, #72309489, #72309501, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35789);