Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 | #37379 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1533 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)(17,18,22,27)(19,28,23,24)(20,31,33,44)(21,34,36,26)(25,38,41,29)(30,39,37,43)(32,48,35,47)(40,45,42,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,38)(21,30,36,37)(25,31,41,44)(26,39,34,43)(32,45,35,46)(40,47,42,48);(3,8)(5,7)(10,13)(11,15)(17,20,22,33)(18,25,27,41)(19,29,23,38)(21,35,36,32)(24,31,28,44)(26,42,34,40)(30,46,37,45)(39,48,43,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,36)(18,26,27,34)(19,30,23,37)(20,32,33,35)(24,39,28,43)(25,40,41,42)(29,45,38,46)(31,47,44,48);(10,13)(14,16)(21,36)(26,34)(30,37)(32,35)(39,43)(40,42)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6837681);