Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1330 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,30,33,24)(20,34,37,25)(21,29,38,26)(22,27,32,40)(31,39,48,45)(35,41,47,43)(36,42,46,44);(9,11)(10,13)(12,16)(14,15)(17,19,23,33)(18,24,28,30)(20,35,37,47)(21,32,38,22)(25,41,34,43)(26,40,29,27)(31,46,48,36)(39,42,45,44);(2,6)(7,8)(10,14)(11,16)(17,20,21,36,23,37,38,46)(18,25,29,44,28,34,26,42)(19,31,22,35,33,48,32,47)(24,39,40,41,30,45,27,43);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,38)(18,26,28,29)(19,32,33,22)(20,36,37,46)(24,40,30,27)(25,42,34,44)(31,47,48,35)(39,41,45,43);(1,4,5,3)(2,8,6,7)(17,22,38,19,23,32,21,33)(18,27,26,30,28,40,29,24)(20,35,46,31,37,47,36,48)(25,43,42,45,34,41,44,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6748911);