Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C2 | C2×#1790 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2404 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)(3,12,15,8)(4,9)(5,10,19,20)(6,11)(7,14)(13,26,28,23)(16,24,30,31)(17,25)(18,22)(21,27)(29,32);(1,3)(2,7)(4,13)(5,14)(6,15)(8,20)(9,21)(10,12)(11,22)(16,27)(17,28)(18,19)(23,31)(24,26)(25,32)(29,30);(1,3)(2,8)(4,13)(5,18)(6,15)(7,20)(9,23)(10,22)(11,12)(14,19)(16,29)(17,28)(21,31)(24,32)(25,26)(27,30);(1,4,6,17)(2,9,11,25)(3,13,15,28)(5,16,19,30)(7,21,22,32)(8,23,12,26)(10,24,20,31)(14,27,18,29)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,f^b=fhi,g^b=gh,e^c=eh,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261398, #72261408, #72261508, … |
| maximal normal subgroup of | #72261398, #72261408, #72261508, … |
| maximal quotient of | #72285458, #72303382, #72303828 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400789);