Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25)(19,27,30,23)(20,24,33,40)(22,26,35,42)(28,38,31,36)(29,39,32,37)(34,41,45,48)(43,47,44,46);(1,2)(3,6)(4,8)(5,7)(17,19,21,30)(18,23,25,27)(20,32,33,29)(22,28,35,31)(24,39,40,37)(26,36,42,38)(34,44,45,43)(41,47,48,46);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(19,28)(22,35)(23,36)(26,42)(27,38)(29,43)(30,31)(32,44)(34,45)(37,46)(39,47)(41,48);(2,7)(6,8)(10,13)(14,16)(17,20,21,33)(18,24,25,40)(19,29,30,32)(22,34,35,45)(23,37,27,39)(26,41,42,48)(28,43,31,44)(36,46,38,47);(1,4,5,3)(2,8,7,6)(9,11,12,15)(10,14,13,16)(20,33)(24,40)(29,32)(34,45)(37,39)(41,48)(43,44)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fi,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7420227);