Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1447 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26)(27,32)(28,38)(29,34)(30,44)(31,35)(33,41)(36,47)(37,45)(39,50)(40,51)(42,48)(43,54)(46,52)(49,55)(53,56);(1,2)(3,7)(4,6)(5,8)(10,14)(12,19)(15,20)(18,23)(25,27,31,30)(26,32,35,44)(28,36,40,41)(29,37,42,43)(33,38,47,51)(34,45,48,54)(39,49,53,52)(46,50,55,56);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,41,31,40,27,36)(26,33,32,38,35,47,44,51)(29,39,43,52,42,53,37,49)(34,46,45,50,48,55,54,56);(3,8)(5,7)(10,14)(11,17)(16,21)(18,23)(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);(2,6)(5,7)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(28,40)(29,42)(33,47)(34,48)(36,41)(37,43)(38,51)(45,54)Polycyclic presentation:
2^9|b^2=i,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819085);