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×#1655 | 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 | |
590 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 590 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,10)(8,16,14,18)(9,12,15,19)(11,13,20,17);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,18)(14,20)(15,19)(16,17);(7,13)(8,17)(10,18)(11,16)(14,15)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=egh,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,e^c=ehi,f^c=fg,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59771);