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 | #29675 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2785 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)(21,22,27,40,26,30,25,31)(23,32,34,28,36,44,35,43)(24,37,41,45,38,29,39,52)(33,47,49,51,48,46,50,42);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23)(22,28)(24,33)(25,34)(26,36)(27,35)(29,42)(30,43)(31,44)(32,40)(37,51)(38,48)(39,49)(41,50)(45,47)(46,52);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,41)(28,42)(30,37)(31,45)(32,46)(34,49)(35,50)(36,48)(40,52)(43,51)(44,47);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,30)(24,38)(28,43)(31,40)(32,44)(33,48)(39,41)(49,50);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,26,27)(22,31,30,40)(23,34,36,35)(24,39,38,41)(28,44,43,32)(29,45,37,52)(33,49,48,50)(42,47,51,46)Polycyclic presentation:
2^9|a^2=fg,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451359);