Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4872 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,6)(3,7,8,4)(10,14)(11,15)(18,23)(21,34)(24,28)(25,31)(30,40)(32,45)(36,41)(43,47);(9,10,13,14)(11,12,15,16)(18,23)(19,27)(25,31)(26,35)(29,39)(30,40)(38,44)(43,47);(1,3,5,8)(2,4,6,7)(9,11)(10,12)(13,15)(14,16)(17,18,22,23)(19,28,27,24)(20,31,33,25)(21,26,34,35)(29,36,39,41)(30,42,40,37)(32,44,45,38)(43,46,47,48);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,24)(20,29)(21,30)(22,27)(23,28)(25,36)(26,37)(31,41)(32,43)(33,39)(34,40)(35,42)(38,46)(44,48)(45,47);(17,20,22,33)(18,25,23,31)(19,29,27,39)(21,32,34,45)(24,36,28,41)(26,38,35,44)(30,43,40,47)(37,46,42,48);(17,21,22,34)(18,26,23,35)(19,30,27,40)(20,32,33,45)(24,37,28,42)(25,38,31,44)(29,43,39,47)(36,46,41,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581833);