Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125647 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| 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 | |
559 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 559 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:
(1,2)(3,5)(4,7)(6,8)(10,15)(11,19)(12,13)(16,23)(17,20)(18,24)(21,22);(2,5)(3,8)(6,7)(9,10)(11,20)(12,17)(13,16)(14,18)(15,19)(21,24)(22,23);(11,21)(12,13)(15,23)(16,17)(19,22)(20,24)Polycyclic presentation:
2^9|d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gi,d^b=df,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72213198, C2×#125647, #72217788, … |
| maximal normal subgroup of | C2×#125647, #72217788, #72217793, … |
| maximal quotient of | #72213160, #72213171, #72213174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 32843);