Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1511 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,24)(20,30)(21,26)(22,35)(23,27)(25,28)(29,38)(31,43)(32,37)(33,41)(34,45)(36,40)(39,42)(44,47)(46,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,29,33,34)(25,37,40,30)(26,38,41,45)(31,42,44,46)(39,47,48,43);(3,8)(5,7)(10,13)(14,16)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,33,44)(24,37,35,30)(26,39,41,48)(29,42,34,46)(38,47,45,43);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,33)(18,26,27,41)(19,29,22,34)(20,31,32,44)(24,38,35,45)(25,39,40,48)(28,42,36,46)(30,43,37,47);(10,13)(14,16)(21,33)(26,41)(29,34)(31,44)(38,45)(39,48)(42,46)(43,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678318);