Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ 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)(25,26,31,43,30,35,29,34)(27,32,44,54,37,46,38,47)(28,39,40,33,42,48,41,49)(36,50,52,45,51,56,53,55);(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)(28,36)(29,38)(30,37)(31,44)(33,45)(34,47)(35,46)(39,50)(40,52)(41,53)(42,51)(43,54)(48,56)(49,55);(2,5)(3,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,30,42)(26,33,35,49)(27,36,37,51)(29,40,31,41)(32,45,46,55)(34,48,43,39)(38,52,44,53)(47,56,54,50);(1,3,4,7)(2,6,5,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(27,37)(32,46)(36,51)(38,44)(45,55)(47,54)(50,56)(52,53);(2,5)(6,8)(10,14)(15,21)(16,22)(23,24)(25,29,30,31)(26,34,35,43)(27,38,37,44)(28,40,42,41)(32,47,46,54)(33,48,49,39)(36,52,51,53)(45,56,55,50)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=fi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,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, 6844378);