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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1265 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)(3,4)(5,7)(6,8)(9,10,13,15)(11,14,12,16)(17,18,22,26)(19,23,30,37)(20,24,31,38)(21,25,33,40)(27,34,29,36)(28,35,42,46)(32,39,44,48)(41,45,43,47);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,28)(21,27)(22,30)(24,35)(25,34)(26,37)(29,33)(31,42)(32,41)(36,40)(38,46)(39,45)(43,44)(47,48);(9,12,13,11)(10,14,15,16)(19,27)(21,33)(23,34)(25,40)(28,41)(29,30)(32,44)(35,45)(36,37)(39,48)(42,43)(46,47);(2,7)(3,8)(10,15)(14,16)(17,20,22,31)(18,24,26,38)(19,28,30,42)(21,32,33,44)(23,35,37,46)(25,39,40,48)(27,41,29,43)(34,45,36,47);(1,4,5,6)(2,3,7,8)(20,31)(24,38)(28,42)(32,44)(35,46)(39,48)(41,43)(45,47)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7417283);