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 | #35737 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 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)(11,14)(12,16)(13,15)(17,18)(19,23)(20,25)(21,24)(22,26)(27,43)(28,37)(29,38)(30,36)(31,41)(32,39)(33,42)(34,40)(35,44)(45,48)(46,47);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,31)(21,28)(22,27)(24,37)(25,41)(26,43)(29,33)(30,34)(32,46)(35,45)(36,40)(38,42)(39,47)(44,48);(9,12,13,11)(10,14,15,16)(18,24)(19,27)(22,34)(23,36)(25,39)(28,30)(29,31)(33,35)(37,43)(38,47)(41,48)(45,46);(2,6)(5,7)(19,28)(20,32)(23,37)(25,39)(27,30)(33,35)(36,43)(42,44);(3,8)(5,7)(17,20,21,32)(18,25,24,39)(19,29,28,45)(22,35,34,33)(23,38,37,48)(26,44,40,42)(27,31,30,46)(36,47,43,41)Polycyclic presentation:
2^9|b^2=fgi,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597706);