Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ 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:
(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26)(27,31)(28,33)(29,42)(30,35)(32,41)(34,44)(36,52)(37,46)(38,47)(39,56)(40,51)(43,50)(45,53)(48,55)(49,54);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,30,38)(26,31,35,47)(28,37,40,36)(29,39,44,55)(32,49,50,45)(33,46,51,52)(34,48,42,56)(41,54,43,53);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,28,30,40)(26,32,35,50)(27,36,38,37)(29,43,44,41)(31,45,47,49)(33,34,51,42)(39,54,55,53)(46,56,52,48);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28)(26,33)(27,37)(29,41)(30,40)(31,46)(32,42)(34,50)(35,51)(36,38)(39,54)(43,44)(45,48)(47,52)(49,56)(53,55);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(28,40)(31,47)(32,50)(33,51)(39,55)(41,43)(48,56)Polycyclic presentation:
2^9|b^2=fi,c^2=i,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,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6796977);