Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9974 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1438 |
| prime radical | #60734 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1190 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 1190 conjugacy classes of subgroups — too many to draw. Only its 142 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:
(5,8)(9,10,14,17,15,19,13,18)(11,20,21,12,22,23,16,24);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,21)(15,22)(17,23)(18,20)(19,24);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,17)(13,20)(14,23)(15,24)(18,22)(19,21)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313861, #72313884, #72313901 |
| maximal normal subgroup of | #72313861, #72313884, #72313901 |
| maximal quotient of | #72314179, #72314219 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18670);