Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1611 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)(15,19)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(37,38,42,45)(39,43,47,40)(41,44,50,52)(46,51,49,48);(8,16)(12,17)(15,19)(18,20)(23,29)(26,33)(30,35)(34,36)(37,39,42,47)(38,40,45,43)(41,49,50,46)(44,51,52,48);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23,25,29)(22,26,28,33)(24,30,32,35)(27,34,31,36)(37,39)(38,43)(40,45)(41,49)(42,47)(44,48)(46,50)(51,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,40,42,43)(38,39,45,47)(41,48,50,51)(44,46,52,49);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41)(38,44)(39,46)(40,48)(42,50)(43,51)(45,52)(47,49)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fgh,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389177);