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)(27,31,38,45)(28,39,41,32)(29,33,36,43)(35,42,48,54)(37,50,51,44)(40,52,49,46)(47,55,56,53);(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)(28,40)(29,36)(31,42)(32,46)(33,43)(37,51)(38,48)(39,52)(41,49)(44,50)(45,54);(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)(26,31,43,54,34,45,33,42)(28,37,49,56,41,51,40,47)(32,44,52,55,39,50,46,53);(10,16)(14,17)(15,22)(21,23)(28,41)(32,39)(37,51)(40,49)(44,50)(46,52)(47,56)(53,55);(11,19)(14,17)(18,24)(21,23)(25,28,36,49,30,41,29,40)(26,32,43,52,34,39,33,46)(27,37,48,56,38,51,35,47)(31,44,54,55,45,50,42,53)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,e^a=ei,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, 6719760);