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)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(38,39)(40,47)(41,49)(50,51);(7,13)(11,16)(14,19)(18,20)(23,29)(26,33)(30,35)(34,36)(37,38,42,39)(40,45,47,43)(41,44,49,46)(48,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,40)(38,43)(39,45)(41,48)(42,47)(44,50)(46,51)(49,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41,42,49)(38,44,39,46)(40,48,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=fg,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, 4511210);