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)(19,23)(20,25)(21,24)(22,26)(27,40)(28,35)(29,36)(30,34)(31,37)(32,39)(33,38)(41,48)(42,46)(43,45)(44,47);(1,2)(3,6)(4,8)(5,7)(17,19,21,29)(18,23,24,36)(20,28,31,42)(22,27,33,30)(25,35,37,46)(26,40,38,34)(32,41,44,43)(39,48,47,45);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(18,24)(19,27)(22,33)(23,34)(25,37)(28,41)(29,30)(32,44)(35,45)(36,40)(42,43)(46,48);(2,7)(6,8)(10,13)(14,16)(17,20,21,31)(18,25,24,37)(19,28,29,42)(22,32,33,44)(23,35,36,46)(26,39,38,47)(27,41,30,43)(34,45,40,48);(1,4,5,3)(2,8,7,6)(9,11,12,15)(10,14,13,16)(20,31)(25,37)(28,42)(32,44)(35,46)(39,47)(41,43)(45,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,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, 7420223);