Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13429 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2044 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #11174 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
769 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,15,16)(4,5,18,21)(6,9,24,27)(7,10)(8,13)(12,30)(14,29)(17,23)(19,26)(20,22)(25,32,31,28);(1,3)(2,8)(4,16)(5,19)(6,14)(7,15)(9,25)(10,26)(11,18)(12,17)(13,21)(20,32)(22,29)(23,28)(24,30)(27,31);(1,4,5,10,7,18,21,2)(3,12,13,25,15,29,19,31)(6,22,20,9,24,17,23,27)(8,14,11,28,26,30,16,32)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,e^c=eh,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^e=gi,g^f=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55673);