Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490472 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #3147 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
436 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,23,27)(19,28,32,45)(20,25,22,37)(21,35,36,48)(24,38,41,31)(26,34,42,29)(30,43,33,44)(39,47,40,46);(2,6)(7,15)(8,13)(9,16)(17,19,29,46,22,30,36,38,23,32,34,47,20,33,21,31)(18,24,35,44,25,40,42,45,27,41,48,43,37,39,26,28);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9);(17,20,23,22)(18,25,27,37)(19,30,32,33)(21,34,36,29)(24,39,41,40)(26,35,42,48)(28,43,45,44)(31,46,38,47)Polycyclic presentation:
2^9|a^2=i,b^2=egh,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,g^a=gi,d^b=di,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397829);