Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2022 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,26,30)(23,31,33,47)(24,28,36,44)(25,38,39,29)(27,40,42,34)(32,46,35,50)(37,52,48,45)(41,51,43,49);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,34,26,33,39,40)(22,27,29,31,30,42,38,47)(24,35,37,51,36,32,48,49)(28,43,45,50,44,41,52,46);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24,26,36)(22,28,30,44)(23,32,33,35)(25,37,39,48)(27,41,42,43)(29,45,38,52)(31,46,47,50)(34,49,40,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(24,36)(28,44)(32,35)(37,48)(41,43)(45,52)(46,50)(49,51);(8,15)(11,18)(14,19)(17,20)(23,33)(27,42)(31,47)(32,35)(34,40)(41,43)(46,50)(49,51)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6477021);