Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| 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 | |
450 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)(17,18,22,26,21,28,36,45,23,27,38,44,37,25,41,43)(19,29,32,35,33,40,47,34,30,42,46,48,31,24,39,20);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,26)(21,33)(22,32)(23,30)(25,42)(27,40)(28,29)(31,37)(34,44)(35,45)(36,47)(38,46)(39,41)(43,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,29,21,35,36,40,23,34,38,42,37,48,41,24)(18,19,26,32,28,33,45,47,27,30,44,46,25,31,43,39);(2,8)(3,10)(6,9)(11,16)(19,30)(20,34)(24,40)(29,42)(31,33)(32,46)(35,48)(39,47)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,d^a=dgi,e^a=eg,d^b=di,e^b=eg,d^c=dg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72320219, #72320234, #72320272, … |
| maximal normal subgroup of | #72320219, #72320234, #72320272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390907);