Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C8 | C24 |
| commutator, socle | C22 | C22×C4×C8 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
542 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,25)(21,26)(22,24)(23,30)(28,39)(29,40)(31,38)(32,36)(33,37)(34,42)(35,43)(41,47)(44,46)(45,48);(17,19,20,28,21,29,31,41,22,30,32,42,33,43,44,48)(18,23,25,34,26,35,38,45,24,27,36,39,37,40,46,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(18,24)(23,27)(25,36)(26,37)(34,39)(35,40)(38,46)(45,47);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bi,c^a=ci,d^c=dg200 conjugacy classes — view the full character table.
| maximal subgroup of | #72231856, #72231860, #72231904, … |
| maximal normal subgroup of | #72231856, #72231860, #72231904, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387140);