Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| 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×D4 | C22×C4 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1867 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,16,11,15,27,32)(7,13,18,30,17,28,35,36)(8,19,20,33,21,12,25,29)(9,22,24,34,23,14,26,31);(1,2)(3,4)(5,7,9,8)(6,12,14,28)(10,18,26,25)(11,17,23,21)(13,15,19,22)(16,33,34,36)(20,27,35,24)(29,31,30,32);(2,4)(5,8,9,17,11,21,23,7)(6,13,14,19,15,28,22,12)(10,25,26,35,27,20,24,18)(16,30,34,29,32,36,31,33)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=i,g^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=eh,f^c=fhi,g^c=gh,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, 56228);