Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 3 generators on 24 points:
(5,8)(10,15,17,24)(11,19)(12,13)(16,21,18,23)(20,22);(1,2)(3,5)(4,6)(7,8)(11,20)(12,13)(15,21)(16,18)(19,22)(23,24);(1,3,4,7)(2,5,6,8)(9,10)(11,21)(12,16)(13,18)(14,17)(15,20)(19,24)(22,23)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=eg,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72217793, #72219624, #72226641 |
| maximal normal subgroup of | #72217793, #72219624, #72226641 |
| maximal quotient of | #72214402, #72214436, #72214443, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33143);