Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1371 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)(3,5)(4,6)(7,8)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26,30,34)(27,35,38,31)(28,39,42,46)(29,43,44,33)(32,40,48,36)(37,45,51,49)(41,54,55,52)(47,50,56,53);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,30,38)(26,31,34,35)(28,40,42,36)(29,37,44,51)(32,46,48,39)(33,45,43,49)(41,53,55,50)(47,54,56,52);(2,6)(3,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,27,36,30,42,38,40)(26,32,31,39,34,48,35,46)(29,41,37,50,44,55,51,53)(33,47,45,52,43,56,49,54);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29)(26,33)(27,37)(28,41)(30,44)(31,45)(32,47)(34,43)(35,49)(36,50)(38,51)(39,52)(40,53)(42,55)(46,54)(48,56);(3,7)(5,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(29,44)(33,43)(37,51)(41,55)(45,49)(47,56)(50,53)(52,54)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgh,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6673211);