Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1217 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,34)(27,31,38,44)(28,32,40,46)(29,33,36,42)(35,41,48,52)(37,43,50,54)(39,45,49,53)(47,51,55,56);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(27,35)(28,39)(29,36)(31,41)(32,45)(33,42)(37,50)(38,48)(40,49)(43,54)(44,52)(46,53);(2,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,36,48,30,38,29,35)(26,31,42,52,34,44,33,41)(28,37,49,55,40,50,39,47)(32,43,53,56,46,54,45,51);(10,16)(14,17)(15,22)(21,23)(28,40)(32,46)(37,50)(39,49)(43,54)(45,53)(47,55)(51,56);(11,19)(14,17)(18,24)(21,23)(25,28,29,39,30,40,36,49)(26,32,33,45,34,46,42,53)(27,37,35,47,38,50,48,55)(31,43,41,51,44,54,52,56)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fhi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719695);