Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7394 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1343 |
| commutator | C2×C22⋊C4 | C2×C8 |
| Frattini, cosocle | C22⋊#95 | C22 |
| prime radical | C2×#534 | C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C25 ◁ C22⋊#95 ◁ G | |
| LECS | G ▷ C22⋊#95 ▷ C25 ▷ C22 ▷ C2 ▷ C1 | |
568 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 568 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,15)(11,17,19,22,20,23,18,14)(13,16,21,24);(2,5)(3,8)(6,7)(9,11)(10,14)(12,18)(13,20)(15,23)(16,22)(17,24)(19,21)Polycyclic presentation:
2^9|a^2=d,c^2=f,d^2=g,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=def,g^b=gi,h^b=hi,d^c=dh,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72226647, #72238059 |
| maximal normal subgroup of | #72226647, #72238059 |
| maximal quotient of | #72226645, #72238067, #72267802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1097);