Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #361285 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2290 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13899 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13)(3,9)(4,17)(5,11,7,12)(8,21)(10,20)(14,28,30,22)(15,23)(16,25)(18,27)(19,26)(24,29,32,31);(1,3,7,21,6,16,5,15)(2,8,11,9,13,23,12,25)(4,14,20,31,19,30,18,29)(10,22,17,24,27,28,26,32);(1,3)(2,9)(4,14)(5,15)(6,16)(7,21)(8,12)(10,24)(11,23)(13,25)(17,28)(18,29)(19,30)(20,31)(22,26)(27,32);(1,4,7,20,6,19,5,18)(2,10,11,17,13,27,12,26)(3,14,21,31,16,30,15,29)(8,22,9,24,23,28,25,32)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319403 |
| maximal normal subgroup of | #72319403 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268649);