Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13357 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C22×Q8 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
624 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,12,18)(7,19)(8,24)(9,15)(10,16)(11,29,28,17)(13,25,14,20)(21,35,36,32)(22,31)(23,34)(26,33)(27,30);(1,2)(3,4)(5,7,10,21,12,23,26,36)(6,13,16,31,18,14,33,24)(8,9,25,11,22,27,20,28)(15,19,29,32,30,34,17,35);(2,4)(5,8,9,21,12,22,27,36)(6,14,15,34,18,13,30,19)(7,10,20,28,23,26,25,11)(16,24,17,32,33,31,29,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,f^b=fh,g^b=ghi,h^b=hi,e^c=ehi,f^c=fh,g^c=ghi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,f^e=fi,g^e=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56220);