Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 | #35210 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1535 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)(19,28)(20,32)(21,27)(22,26)(23,36)(24,30)(25,33)(29,44)(31,40)(34,42)(35,39)(37,41)(38,46)(43,47)(45,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,38)(26,28)(27,40)(32,48)(33,47)(34,45)(36,39)(37,46)(41,43)(42,44);(3,8)(5,7)(10,13)(14,16)(17,20,22,34)(18,25,26,41)(19,29,30,45)(21,33,23,37)(24,38,28,43)(27,42,36,32)(31,46,35,47)(39,48,40,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(25,41)(27,36)(29,45)(31,35)(32,42)(46,47);(10,13)(14,16)(17,21,22,23)(18,27,26,36)(19,31,30,35)(20,33,34,37)(24,39,28,40)(25,42,41,32)(29,46,45,47)(38,48,43,44)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678551);