Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632048 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1386 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)(4,6)(5,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,26,30,34)(27,37,39,32)(28,41,43,50)(29,36,31,35)(33,45,48,42)(38,52,53,56)(40,47,44,49)(46,55,51,54);(1,3,4,8)(2,5,6,7)(9,11,13,19)(10,14,16,17)(12,21,22,18)(15,24,20,23)(25,27,30,39)(26,32,34,37)(28,38,43,53)(29,44,31,40)(33,46,48,51)(35,49,36,47)(41,56,50,52)(42,55,45,54);(3,8)(5,7)(25,28)(26,33)(27,38)(29,42)(30,43)(31,45)(32,46)(34,48)(35,41)(36,50)(37,51)(39,53)(40,54)(44,55)(47,52)(49,56);(2,6)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(26,34)(27,39)(33,48)(35,36)(38,53)(40,44)(41,50)(54,55);(10,16)(14,17)(15,20)(23,24)(25,29,30,31)(26,35,34,36)(27,40,39,44)(28,42,43,45)(32,47,37,49)(33,41,48,50)(38,54,53,55)(46,52,51,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6551919);