Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | |
2824 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)(21,22,27,31)(23,32,35,42)(24,29,37,45)(25,30,26,39)(28,40,43,34)(33,47,49,52)(36,44,38,46)(41,48,51,50);(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,23)(22,28)(24,33)(25,34)(26,40)(27,35)(29,41)(30,42)(31,43)(32,39)(36,48)(37,49)(38,50)(44,47)(45,51)(46,52);(2,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,24)(22,29)(23,33)(25,38)(26,36)(27,37)(28,41)(30,46)(31,45)(32,47)(34,50)(35,49)(39,44)(40,48)(42,52)(43,51);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,25,27,26)(22,30,31,39)(23,34,35,40)(24,36,37,38)(28,42,43,32)(29,44,45,46)(33,48,49,50)(41,47,51,52);(8,16)(11,18)(14,20)(17,19)(24,37)(29,45)(33,49)(36,38)(41,51)(44,46)(47,52)(48,50)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6478856);