Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343418 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62816 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,27,30,23)(20,31)(21,25)(22,26)(24,33)(28,35,42,39)(29,40,43,36)(32,44)(34,38)(37,45)(41,46,48,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,23)(20,28)(21,29)(22,30)(24,35)(25,36)(26,27)(31,39)(32,41)(33,42)(34,43)(37,46)(38,40)(44,47)(45,48);(2,7)(6,14)(8,15)(9,16)(17,20,22,33)(18,24,26,31)(19,28,30,42)(21,32,34,45)(23,35,27,39)(25,37,38,44)(29,41,43,48)(36,46,40,47);(17,21)(18,25)(19,29)(20,32)(22,34)(23,36)(24,37)(26,38)(27,40)(28,41)(30,43)(31,44)(33,45)(35,46)(39,47)(42,48);(21,34)(25,38)(29,43)(32,45)(36,40)(37,44)(41,48)(46,47)Polycyclic presentation:
2^9|a^2=g,c^2=h,f^2=i,b^a=bf,c^a=chi,f^a=fh,f^b=fi,g^b=ghi,e^d=eh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250616);