Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632048 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
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)(25,26,30,35,31,36,29,34)(27,32,40,48,39,47,38,46)(28,41,42,49,43,50,44,33)(37,51,52,55,53,56,54,45);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,32)(28,37)(29,38)(30,40)(31,39)(33,45)(34,46)(35,48)(36,47)(41,51)(42,52)(43,53)(44,54)(49,55)(50,56);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,45)(34,41)(35,50)(36,49)(38,52)(39,53)(40,54)(46,51)(47,55)(48,56);(3,7)(5,8)(10,16)(11,19)(15,22)(18,24)(25,29,31,30)(26,34,36,35)(27,38,39,40)(28,42,43,44)(32,46,47,48)(33,41,49,50)(37,52,53,54)(45,51,55,56);(2,6)(5,8)(10,16)(14,17)(15,22)(21,23)(27,39)(32,47)(37,53)(38,40)(45,55)(46,48)(51,56)(52,54)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,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, 6542028);