Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1468 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)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,21)(19,24,30,28)(20,32,34,25)(23,26,37,36)(27,45,46,39)(29,40,31,41)(33,48,47,42)(35,44,38,43);(1,3,4,8)(2,5,6,7)(17,19)(18,24)(20,33)(21,28)(22,30)(23,29)(25,42)(26,40)(27,35)(31,37)(32,48)(34,47)(36,41)(38,46)(39,43)(44,45);(10,14)(15,16)(17,20,22,34)(18,25,21,32)(19,27,30,46)(23,38,37,35)(24,39,28,45)(26,44,36,43)(29,33,31,47)(40,42,41,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,21,22,18)(19,28,30,24)(20,25,34,32)(23,36,37,26)(27,39,46,45)(29,41,31,40)(33,42,47,48)(35,43,38,44);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,29)(23,37)(24,40)(26,36)(27,33)(28,41)(30,31)(35,38)(39,42)(43,44)(45,48)(46,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=g,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6637429);