Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141417 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1964 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | #2408 | C22 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
479 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 479 conjugacy classes of subgroups — too many to draw. Only its 80 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 — 3 generators on 24 points:
(2,3)(4,5)(7,8)(9,10)(11,17)(12,15,21,22)(13,16)(14,19,23,18)(20,24);(1,2,5,7,6,8,4,3)(9,11,12,18,13,20,21,19)(10,14,22,24,16,23,15,17);(1,3,4,8,6,7,5,2)(9,12,13,21)(10,15,16,22)(11,18,20,19)(14,17,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=egi,d^2=ghi,e^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48661);