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 | #38035 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,21,27,22,26,33,24)(19,23,30,34,28,36,42,37)(20,25,32,39,31,40,44,38)(29,35,43,45,41,46,48,47);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15)(17,19,22,28)(18,23,26,36)(20,29,31,41)(21,30,33,42)(24,37,27,34)(25,35,40,46)(32,43,44,48)(38,47,39,45);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(18,24)(19,28)(21,33)(23,34)(25,38)(26,27)(29,41)(32,44)(35,45)(36,37)(39,40)(46,47);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20,22,31)(18,25,26,40)(19,29,28,41)(21,32,33,44)(23,35,36,46)(24,38,27,39)(30,43,42,48)(34,45,37,47);(1,4,5,3)(2,6,7,8)(18,26)(19,28)(20,31)(24,27)(30,42)(32,44)(35,46)(45,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=f,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6782318);