Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #38903 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(21,22,26,29)(23,31,34,44)(24,28,35,42)(25,37,39,30)(27,38,41,33)(32,45,46,52)(36,49,48,43)(40,50,51,47);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23,26,34)(22,27,29,41)(24,32,35,46)(25,38,39,33)(28,40,42,51)(30,44,37,31)(36,50,48,47)(43,52,49,45);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24)(22,28)(23,32)(25,36)(26,35)(27,40)(29,42)(30,43)(31,45)(33,47)(34,46)(37,49)(38,50)(39,48)(41,51)(44,52);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(22,29)(24,35)(27,41)(30,37)(31,44)(32,46)(36,48)(47,50);(8,16)(11,18)(14,20)(17,19)(24,35)(28,42)(32,46)(36,48)(40,51)(43,49)(45,52)(47,50)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6478780);