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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(10,14)(12,19)(15,21)(17,20)(25,26,31,37,30,36,29,35)(27,38,40,51,41,32,45,49)(28,42,43,50,44,34,46,33)(39,48,56,47,55,53,54,52);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27)(26,32)(28,39)(29,45)(30,41)(31,40)(33,47)(34,48)(35,51)(36,38)(37,49)(42,53)(43,56)(44,55)(46,54)(50,52);(2,5)(3,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,30,44)(26,33,36,50)(27,39,41,55)(29,43,31,46)(32,47,38,52)(34,35,42,37)(40,54,45,56)(48,51,53,49);(1,3,4,7)(2,6,5,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,28)(26,34)(27,39)(29,46)(30,44)(31,43)(32,48)(33,37)(35,50)(36,42)(38,53)(40,56)(41,55)(45,54)(47,49)(51,52);(2,5)(6,8)(10,14)(15,21)(16,22)(23,24)(25,29,30,31)(26,35,36,37)(27,40,41,45)(28,43,44,46)(32,49,38,51)(33,42,50,34)(39,54,55,56)(47,48,52,53)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,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, 6844304);