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 | C2×#447 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1513 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,14)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,33)(28,38)(29,43)(32,40)(34,44)(35,42)(36,39)(37,41)(45,46)(47,48);(1,3,4,8)(2,5,6,7)(17,19)(18,23)(20,32)(21,30)(22,36)(24,40)(25,27)(26,43)(28,34)(29,35)(31,48)(33,47)(37,46)(38,41)(39,42)(44,45);(10,14)(15,16)(17,20)(18,24)(19,28)(21,33)(22,37)(23,38)(25,31)(26,44)(27,45)(29,47)(30,46)(32,36)(34,35)(39,48)(40,43)(41,42);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,25)(20,33)(23,27)(26,42)(28,46)(32,47)(34,37)(39,43);(3,8)(5,7)(9,12,13,11)(10,16,14,15)(19,29)(22,35)(23,39)(26,42)(27,43)(28,47)(30,36)(32,46)(34,37)(38,48)(40,45)(41,44)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,d^2=gi,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, 6637947);