Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,26,22,28,36,45,23,27,37,44,20,25,34,42)(19,29,32,46,33,24,38,40,30,41,35,48,31,39,47,43);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,24,27,39)(20,31)(21,35)(22,33)(23,30)(25,29,28,41)(26,43,44,40)(32,37)(34,38)(36,47)(42,48,45,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,30)(24,39)(29,41)(31,33)(32,35)(38,47)(40,43)(46,48)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fi,c^b=ci,e^b=egi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72293221, #72293224, #72293282, … |
| maximal normal subgroup of | #72293221, #72293224, #72293282, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34127);