Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | |
1221 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)(27,31,37,44)(28,32,41,48)(29,33,36,43)(35,42,49,53)(38,45,40,47)(39,46,51,55)(50,54,52,56);(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)(31,42)(32,46)(33,43)(37,49)(38,40)(41,51)(44,53)(45,47)(48,55);(2,6)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,27,36,49,30,37,29,35)(26,31,43,53,34,44,33,42)(28,40,51,50,41,38,39,52)(32,47,55,54,48,45,46,56);(10,16)(14,17)(15,22)(21,23)(27,37)(31,44)(35,49)(38,40)(42,53)(45,47)(50,52)(54,56);(11,19)(14,17)(18,24)(21,23)(25,28,36,51,30,41,29,39)(26,32,43,55,34,48,33,46)(27,38,49,52,37,40,35,50)(31,45,53,56,44,47,42,54)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fhi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719607);