Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94176 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1393 |
| commutator | C2×C42 | C42 |
| Frattini, cosocle | C2×#390 | C22 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#390 ◁ G | |
| LECS | G ▷ C2×#390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 431 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)(11,19,20,15)(12,17,13,16)(14,18)(21,24,22,23);(2,5,7,6)(9,11,12,20,14,22,13,21)(10,15,16,19,18,24,17,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fg,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fh,g^a=ghi,h^a=hi,d^b=dg,d^c=dhi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1373);