Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | #127 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | #1438 |
| prime radical | #34547 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1128 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1128 conjugacy classes of subgroups — too many to draw. Only its 146 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:
(1,2,4,7,5,3,8,6)(9,10,13,16,14,17,12,21)(11,18,19,15,20,24,22,23);(1,3)(2,5)(4,6)(7,8)(9,11)(10,15)(12,22)(13,19)(14,20)(16,24)(17,23)(18,21);(1,2)(3,5)(4,7)(6,8)(15,23)(18,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239454, #72257874, #72265312, … |
| maximal normal subgroup of | #72239454, #72257874, #72265312, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 18694);