Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1351 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 1351 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,7)(9,10,14,18,15,11,13,17)(12,19,22,24,23,16,21,20);(1,2,4,5)(3,7,8,6)(9,11,15,10)(12,20,23,24)(13,17,14,18)(16,21,19,22);(9,11)(10,15)(12,19)(13,17)(14,18)(16,23)(20,22)(21,24);(1,3,4,8)(2,6,5,7)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72339032, #72339040, #72339076, … |
| maximal normal subgroup of | #72339032, #72339040, #72339076, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337679);