Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1259 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,19)(13,15)(16,20)(17,21)(18,24)(22,23)(25,26,31,35)(27,32,30,44)(28,38,41,51)(29,34,42,48)(33,36,47,40)(37,45,43,54)(39,50,52,56)(46,49,55,53);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27,31,30)(26,32,35,44)(28,36,41,40)(29,37,42,43)(33,38,47,51)(34,45,48,54)(39,49,52,53)(46,50,55,56);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,19)(18,23)(25,28,30,40,31,41,27,36)(26,33,32,38,35,47,44,51)(29,39,43,53,42,52,37,49)(34,46,45,50,48,55,54,56);(1,4,5,3)(2,8,6,7)(11,17)(14,21)(18,23)(22,24)(25,29)(26,34)(27,37)(28,39)(30,43)(31,42)(32,45)(33,46)(35,48)(36,49)(38,50)(40,53)(41,52)(44,54)(47,55)(51,56);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(29,42)(34,48)(37,43)(39,52)(45,54)(46,55)(49,53)(50,56)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406261);