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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(37,38)(39,44)(40,47)(41,43)(42,45)(46,51)(48,52)(49,50);(6,10)(11,13)(12,16)(18,19)(23,29)(26,33)(30,35)(34,36)(37,39,43,40)(38,44,41,47)(42,46,50,48)(45,51,49,52);(5,6)(7,11)(8,16)(9,10)(12,17)(13,15)(14,19)(18,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35)(37,40,43,39)(38,44,41,47)(42,48,50,46)(45,51,49,52);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,14,17,20)(12,18,16,19)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,41,43,38)(39,44,40,47)(42,49,50,45)(46,51,48,52);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,42,43,50)(38,45,41,49)(39,46,40,48)(44,51,47,52)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=g,d^2=fg,e^2=fgh,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, 4511413);