Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,25,33,40)(19,23,29,36,28,35,42,46)(20,24,32,39,31,38,44,48)(27,34,41,45,30,37,43,47);(1,3,6,2)(4,7,5,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,30)(21,29)(22,28)(24,37)(25,35)(26,36)(27,31)(32,43)(33,42)(34,38)(39,47)(40,46)(41,44)(45,48);(10,14)(13,16)(17,20,21,32,22,31,33,44)(18,24,26,39,25,38,40,48)(19,27,29,41,28,30,42,43)(23,34,36,45,35,37,46,47);(1,4,6,5)(2,7,3,8)(18,25)(23,35)(24,38)(26,40)(34,37)(36,46)(39,48)(45,47);(1,5,6,4)(2,7,3,8)(19,28)(20,31)(23,35)(24,38)(29,42)(32,44)(36,46)(39,48)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=h,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300495);