Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1636 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)(6,10)(7,13)(12,17)(14,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38,42,44)(39,45,47,40)(41,43,50,52)(46,51,49,48);(7,13)(11,16)(14,19)(18,20)(23,29)(27,30)(31,35)(34,36)(37,39,42,47)(38,40,44,45)(41,46,50,49)(43,48,52,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35)(37,39)(38,40)(41,49)(42,47)(43,51)(44,45)(46,50)(48,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,40)(38,39)(41,48)(42,45)(43,46)(44,47)(49,52)(50,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)(37,41)(38,43)(39,46)(40,48)(42,50)(44,52)(45,51)(47,49)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=gh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4513378);