Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | #466 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,23,28,21,27,39,45,24,29,41,46,22,26,40,44)(19,30,33,36,32,25,42,20,34,37,48,35,31,43,47,38);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,25)(20,28)(21,32)(22,31)(23,33)(24,34)(26,30)(27,37)(29,43)(35,45)(36,44)(38,46)(39,42)(40,47)(41,48);(2,8)(7,15)(9,13)(10,16)(17,20,23,37,21,35,39,43,24,38,41,30,22,36,40,25)(18,19,28,33,27,32,45,42,29,34,46,48,26,31,44,47);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,21,24,22)(18,26,29,27)(19,31,34,32)(20,35,38,36)(23,39,41,40)(25,37,43,30)(28,44,46,45)(33,47,48,42)Polycyclic presentation:
2^9|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,d^b=di,e^b=eg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72286483, #72286956, #72287023, … |
| maximal normal subgroup of | #72286483, #72286956, #72287023, … |
| maximal quotient of | #72287193 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390802);