Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,31,36,42)(28,32,40,46)(29,33,39,45)(35,41,47,51)(37,43,49,53)(38,44,50,54)(48,52,55,56);(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(27,35)(28,38)(29,39)(31,41)(32,44)(33,45)(36,47)(37,49)(40,50)(42,51)(43,53)(46,54);(1,3,5,4)(2,6,7,8)(25,27,29,35,30,36,39,47)(26,31,33,41,34,42,45,51)(28,37,38,48,40,49,50,55)(32,43,44,52,46,53,54,56);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,36)(31,42)(35,47)(37,49)(41,51)(43,53)(48,55)(52,56);(1,4,5,3)(2,6,7,8)(11,19)(14,17)(18,24)(22,23)(25,28,39,50,30,40,29,38)(26,32,45,54,34,46,33,44)(27,37,47,55,36,49,35,48)(31,43,51,56,42,53,41,52)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=gh,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6744372);