Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| 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 | #3169 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ #2162 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
449 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,22,29,24,28,41,46,23,27,40,31,21,37,39,47)(19,30,33,36,32,25,42,35,26,44,45,34,38,43,48,20);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,25)(20,31)(21,38)(22,33)(23,26)(24,32)(27,43)(28,44)(29,35)(30,37)(34,46)(36,47)(39,48)(40,45)(41,42);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,30,24,36,41,25,23,35,40,44,21,34,39,43)(18,26,29,45,28,38,46,48,27,19,31,33,37,32,47,42);(2,8)(3,10)(6,9)(11,16)(18,27)(19,26)(28,37)(29,31)(32,38)(33,45)(42,48)(46,47)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dg,e^a=eg,d^b=di,e^b=eg,d^c=dgi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72287023, #72287116, #72287125, … |
| maximal normal subgroup of | #72287023, #72287116, #72287125, … |
| maximal quotient of | #72323196 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390911);