Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(5,6)(7,17)(8,10,14,9)(11,26)(12,16)(13,19,31,20)(15,28)(18,21)(22,32)(23,24,27,25)(29,35)(30,36,33,34);(5,7,10,20)(6,13,8,17)(9,19,12,22)(11,21,24,34)(14,32,16,31)(15,33,27,35)(18,26,30,23)(25,36,28,29);(5,8,10,16,12,14,9,6)(7,18,20,30,22,35,19,33)(11,27,24,26,28,23,25,15)(13,29,32,36,31,21,17,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=di,d^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=fhi,g^b=ghi,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi,e^d=ei,f^d=fi,g^d=gi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56230);