Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 | #23962 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26)(27,32)(28,39)(29,34)(30,44)(31,35)(33,42)(36,53)(37,47)(38,48)(40,56)(41,52)(43,49)(45,51)(46,55)(50,54);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27,31,37)(26,32,35,47)(28,40,41,36)(29,43,30,38)(33,50,51,46)(34,49,44,48)(39,56,52,53)(42,54,45,55);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28,31,41)(26,33,35,51)(27,36,37,40)(29,42,30,45)(32,46,47,50)(34,52,44,39)(38,54,43,55)(48,53,49,56);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(27,37)(28,41)(32,47)(33,51)(38,43)(39,52)(42,45)(48,49);(11,19)(14,17)(18,24)(22,23)(25,29,31,30)(26,34,35,44)(27,38,37,43)(28,42,41,45)(32,48,47,49)(33,52,51,39)(36,54,40,55)(46,53,50,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6824386);