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 | |
1388 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)(25,26,31,35)(27,32,37,46)(28,39,41,49)(29,34,30,43)(33,44,50,40)(36,51,52,56)(38,47,42,48)(45,53,55,54);(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)(26,32)(28,36)(29,42)(30,38)(31,37)(33,45)(34,48)(35,46)(39,51)(40,54)(41,52)(43,47)(44,53)(49,56)(50,55);(3,8)(5,7)(25,28)(26,33)(27,36)(29,40)(30,44)(31,41)(32,45)(34,49)(35,50)(37,52)(38,53)(39,43)(42,54)(46,55)(47,51)(48,56);(2,6)(5,7)(9,12,13,22)(10,15,16,20)(11,18,19,21)(14,23,17,24)(27,37)(32,46)(36,52)(38,42)(45,55)(47,48)(51,56)(53,54);(10,16)(14,17)(15,20)(23,24)(25,29,31,30)(26,34,35,43)(27,38,37,42)(28,40,41,44)(32,47,46,48)(33,49,50,39)(36,53,52,54)(45,51,55,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fh,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=dfhi,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, 6551926);