Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483873 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #466 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,8,5,7,11,14)(3,6,9,13,10,12,15,16)(17,18,21,27,23,28,22,25)(19,29,31,46,33,47,32,43)(20,34,37,44,36,42,38,26)(24,39,40,30,41,35,45,48);(17,19,23,33)(18,24,28,41)(20,35,36,39)(21,32,22,31)(25,40,27,45)(26,43,44,46)(29,34,47,42)(30,38,48,37);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(18,25,28,27)(20,36)(24,40,41,45)(26,34,44,42)(29,43,47,46)(30,48)(35,39)(37,38);(2,7)(6,12)(8,14)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285766, #72286587, #72286619, … |
| maximal normal subgroup of | #72285766, #72286587, #72286619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391075);