Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #363903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2297 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13959 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1234 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)(3,11,14,23)(4,15,17,29)(5,9)(6,10)(7,13,19,28)(8,16,22,30)(12,25)(18,21)(20,27)(24,32)(26,31);(1,3)(2,7,10,19)(4,12)(5,13)(6,14)(8,20,22,25)(9,11,21,23)(15,24,29,31)(16,26)(17,27)(18,28)(30,32);(7,19)(11,23)(20,25)(24,31);(1,4,5,16,6,17,18,30)(2,8,21,29,10,22,9,15)(3,12,13,26,14,27,28,32)(7,20,23,31,19,25,11,24)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=eg,e^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216181, #72216238, … |
| maximal normal subgroup of | #72215972, #72216181, #72216238, … |
| maximal quotient of | #72226702, C2×#364080, #72252134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271276);