Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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×#3326 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 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)(21,22,27,31)(23,32,36,47)(24,29,38,44)(25,30,26,40)(28,34,42,35)(33,46,49,51)(37,43,39,45)(41,50,52,48);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,35,27,36,25,34)(22,28,30,32,31,42,40,47)(24,33,37,48,38,49,39,50)(29,41,45,46,44,52,43,51);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24,27,38)(22,29,31,44)(23,33,36,49)(25,39,26,37)(28,41,42,52)(30,45,40,43)(32,46,47,51)(34,50,35,48);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,38,39)(28,32,42,47)(29,43,44,45)(33,48,49,50)(41,51,52,46);(8,15)(11,18)(14,19)(17,20)(24,38)(29,44)(33,49)(37,39)(41,52)(43,45)(46,51)(48,50)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476889);