Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10386 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1446 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
948 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 948 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:
(1,2)(3,7,8,5)(4,6)(9,10)(11,14,18,24)(12,15)(13,16)(17,21)(19,22,20,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14,16,24)(12,20)(13,18)(15,23,17,22)(19,21);(1,2)(3,5)(4,6)(7,8)(10,15,16,17)(14,22,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=eh,g^b=gh,d^c=dhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72260815, #72260827, #72260848, … |
| maximal normal subgroup of | #72260815, #72260827, #72260848, … |
| maximal quotient of | #72261825, #72303963 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29358);