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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1138 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)(3,6)(4,5)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26)(27,32)(28,33)(29,31)(30,34)(35,47)(36,44)(37,42)(38,46)(39,52)(40,43)(41,48)(45,49)(50,56)(51,54)(53,55);(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(26,31)(27,35)(28,39)(30,36)(32,45)(33,48)(37,49)(40,41)(42,47)(43,52)(46,54)(50,53);(2,5)(6,8)(25,27,36,49,29,37,30,35)(26,32,44,45,31,42,34,47)(28,38,39,53,40,51,41,50)(33,46,52,55,43,54,48,56);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(26,31)(27,37)(33,43)(34,44)(35,49)(38,51)(48,52)(50,53);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,28,30,41,29,40,36,39)(26,33,34,48,31,43,44,52)(27,38,35,50,37,51,49,53)(32,46,47,56,42,54,45,55)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=hi,d^2=fgi,e^2=fhi,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, 6735568);