Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | C2×#2199 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
603 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 603 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,8,11,10,7,14,15)(9,17,19,12,20,13,18,16);(1,2,3,4)(5,7)(6,10)(8,11)(9,17)(12,18)(13,20)(14,15)(16,19);(5,6)(7,10)(8,12)(9,17)(11,18)(13,20)(14,16)(15,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=egi,g^a=gh,h^a=hi,d^b=di,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,f^c=fg,g^c=gi,h^c=hi,e^d=ehi,g^d=gi,f^e=fhi,g^e=gi,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59758);