Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#106 |
| prime radical | C23 | C4×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
249 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 249 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,3,5,6,8,7)(9,10,13,18,14,19,24,22,16,20,23,21,15,11,12,17);(9,11,14,20,16,19,15,10)(12,22,13,18,24,17,23,21);(1,3,4,6,5,7,8,2)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=de,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=cgh,f^a=fi,d^b=dg,e^b=egi,g^b=gi,h^b=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287034, #72287179, #72287192, … |
| maximal normal subgroup of | #72287034, #72287179, #72287192, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36380);