Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1351 |
| commutator | C2×C22⋊C4 | C2×C8 |
| Frattini, cosocle | C22⋊#95 | C22 |
| prime radical | C25 | C22⋊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 | |
424 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 424 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:
(2,5,6,7)(9,10,12,15)(11,17,18,14,19,22,20,23)(13,16,21,24);(1,2,4,6)(3,7,8,5)(9,11,13,19)(10,14)(12,20,21,18)(15,17)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=d,b^2=f,c^2=fi,d^2=g,b^a=bc,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=defi,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 | #72226622, #72238057 |
| maximal normal subgroup of | #72226622, #72238057 |
| maximal quotient of | #72226621, #72238068 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1172);