Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7076 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C22×C8 | C2×C8 |
| Frattini, cosocle | C2×#336 | C22 |
| socle | C22 | #1270 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#336 ◁ G | |
| LECS | G ▷ C2×#336 ▷ C23×C4 ▷ C22 ▷ C1 | |
417 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 50 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)(3,7,8,5)(4,6)(9,10,13,18,15,19,24,17,16,20,12,23,14,11,21,22);(2,5)(6,7)(9,11)(10,15)(12,17)(13,22)(14,20)(16,19)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=d,c^2=fi,d^2=g,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fi,d^c=dhi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47);