Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1185 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,33)(27,35,38,31)(28,32,41,46)(30,34,37,43)(36,47,49,42)(39,48,52,44)(40,45,51,54)(50,55,56,53);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(27,36)(28,40)(30,37)(31,42)(32,45)(34,43)(35,47)(38,49)(39,52)(41,51)(44,48)(46,54);(2,6)(7,8)(25,27,37,49,29,38,30,36)(26,31,43,47,33,35,34,42)(28,39,51,56,41,52,40,50)(32,44,54,55,46,48,45,53);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(27,38)(31,35)(36,49)(39,52)(42,47)(44,48)(50,56)(53,55);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,37,51,29,41,30,40)(26,32,43,54,33,46,34,45)(27,39,49,56,38,52,36,50)(31,44,47,55,35,48,42,53)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=gi,e^2=fgh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750663);