Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #177733 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1622 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1622 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(13,16)(18,22)(19,20)(23,24);(1,2,5,7)(3,6,4,8)(9,10)(11,13)(12,14)(15,16);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,24)(21,23);(2,7)(6,8)(10,14)(13,16)(17,19,21,20)(18,22,23,24)Polycyclic presentation:
2^9|b^2=g,c^2=gh,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,e^c=eh,f^c=fi56 conjugacy classes — view the full character table.
| Fitting subgroup of | #72340891 |
| maximal subgroup of | #72213214, #72213426, #72213442, … |
| maximal normal subgroup of | #72213214, #72213426, #72213442, … |
| maximal quotient of | #72217689, #72217697, #72217742 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 85080);