Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13432 |
| 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 | |
771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,13,8,14,25,10)(3,15,17,23,20,32,31,4)(6,26,27,12,24,11,7,16)(9,29,28,21,18,22,19,30);(1,3)(2,4)(5,17)(6,18)(7,28)(8,20)(9,24)(10,15)(11,30)(12,29)(13,32)(14,23)(16,22)(19,27)(21,26)(25,31);(1,4,5,15)(2,9,10,28)(3,16,17,26)(6,21,27,22)(7,29,24,30)(8,23,25,32)(11,20,12,31)(13,19,14,18)Polycyclic presentation:
2^9|a^2=g,c^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,e^c=ehi,f^c=fi,g^c=ghi,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.
| maximal subgroup of | #72282473 |
| maximal normal subgroup of | #72282473 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55669);