Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491437 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 846 conjugacy classes of subgroups — too many to draw. Only its 206 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:
(2,5)(3,6)(7,8)(10,14)(11,15)(12,18)(16,20)(17,24)(19,23)(21,22);(9,10,15,22,12,17,19,20,13,16,23,24,18,21,11,14);(1,2,6,8,4,7,3,5);(10,16)(14,20)(17,21)(22,24)Polycyclic presentation:
2^9|b^2=eg,c^2=fh,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72213956, #72213983, #72214020, … |
| maximal normal subgroup of | #72213956, #72213983, #72214020, … |
| maximal quotient of | #72213956, #72216165, #72225187, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398665);