Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #268627 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #25747 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1026 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 24 points:
(1,2,4,5,3,6,7,8)(9,10)(11,14)(12,19)(13,15)(16,20)(17,23)(18,22)(21,24);(4,7)(5,8)(10,15)(11,17)(13,23)(16,19)(18,24)(20,22);(1,2)(3,6)(4,8)(5,7)(9,11,14,10)(12,18,21,16)(13,15,23,17)(19,20,24,22);(4,7)(5,8)(9,12)(10,16)(11,18)(13,22)(14,21)(15,19)(17,24)(20,23)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 175893);