Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
3673 conjugacy classes of subgroups; only the 354 normal subgroups are shown.
This group has 3673 conjugacy classes of subgroups — too many to draw. Only its 354 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)(3,4)(9,10)(11,14)(12,15,19,22)(13,16)(17,20,23,24)(18,21);(2,4)(5,6)(7,8)(9,11,13,18)(10,14,16,21)(12,17,19,23)(15,20,22,24);(6,8)(9,12)(10,15,16,22)(11,17)(13,19)(14,20,21,24)(18,23);(2,4)(11,18)(14,21)(17,23)(20,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72251368, #72251411, #72267142, … |
| maximal normal subgroup of | #72251368, #72251411, #72267142, … |
| maximal quotient of | #72255689 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281222);