Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133362 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #2431 | C22 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 78 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)(11,21)(12,23,13,18)(14,19)(15,20)(16,22,17,24);(1,2)(3,5)(4,6)(7,8)(9,11,15,14)(10,16,20,17)(12,22,13,24)(18,19,23,21);(1,3)(2,5)(4,7)(6,8)(9,12)(10,17,20,16)(11,22)(13,15)(14,24)(18,19,23,21)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,f^c=fi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72262574, #72262723, #72262732, … |
| maximal normal subgroup of | #72262574, #72262723, #72262732, … |
| maximal quotient of | #72262576, #72324283, #72324285, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40701);