Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1178 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)(3,7)(4,8)(5,6)(9,10,13,15)(11,17,19,14)(12,16,21,23)(18,24,20,22)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,53)(37,46)(39,56)(40,49)(41,50)(42,51)(43,47)(44,52)(45,54)(48,55);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27,29,38)(26,31,33,35)(28,39,40,55)(30,43,42,37)(32,48,49,56)(34,46,51,47)(36,44,54,41)(45,52,53,50);(2,6)(7,8)(25,28,29,40)(26,32,33,49)(27,36,38,54)(30,44,42,41)(31,45,35,53)(34,52,51,50)(37,55,43,39)(46,48,47,56);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(26,33)(28,40)(31,35)(34,51)(36,54)(39,55)(41,44)(46,47);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(26,33)(27,37)(28,40)(30,42)(31,46)(35,47)(36,39)(38,43)(45,56)(48,53)(50,52)(54,55)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,d^2=g,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6646325);