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 | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
311 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,5,6,4,7,3,8)(9,10,15,19,16,20,21,11)(12,13,22,24,18,14,23,17);(2,6)(3,5)(7,8)(9,11,13,23,15,20,24,18,16,19,14,22,21,10,17,12);(2,7)(6,8)(9,12,13,19,15,23,24,10,16,18,14,11,21,22,17,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=eh,d^2=ghi,e^2=g,f^2=h,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 | #72342429, #72342440, #72342443, … |
| maximal normal subgroup of | #72342429, #72342440, #72342443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48509);