Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480674 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1798 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1798 conjugacy classes of subgroups — too many to draw. Only its 210 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 — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,18,14,19,11,15)(12,17,20,23,22,24,21,16);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,19,18)(12,20,22,21)(16,17,23,24);(3,8)(5,7)(9,11,14,13)(10,16,19,23)(12,21,22,20)(15,17,18,24);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,f^c=fi,g^c=ghi,g^f=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72303600, #72303603 |
| maximal normal subgroup of | #72303600, #72303603 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403406);