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 | #11087 |
| 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 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(3,12,16,15)(9,26,28,32)(10,23)(11,25)(13,31,29,22)(14,30,17,27)(20,24);(1,2,4,8)(3,13,14,26)(5,10,18,23)(6,20,21,24)(7,11,19,25)(9,27,22,15)(12,28,30,31)(16,29,17,32);(1,3)(2,9)(4,17)(5,15)(6,12)(7,16)(8,22)(10,29)(11,28)(13,20)(14,19)(18,30)(21,27)(23,32)(24,26)(25,31)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,g^c=gi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72280714, #72281842, #72282782, … |
| maximal normal subgroup of | #72280714, #72281842, #72282782, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55439);