Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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)(7,13)(8,15)(11,16)(12,17)(21,22)(23,26)(24,29)(25,28)(27,32)(30,33)(31,36)(34,35)(37,38,42,45)(39,43,40,47)(41,44,49,52)(46,50,48,51);(6,10)(11,16)(12,17)(18,20)(22,23)(25,33)(27,29)(31,34)(37,39,42,40)(38,43,45,47)(41,48,49,46)(44,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,36,34,35)(37,40,42,39)(38,43,45,47)(41,48,49,46)(44,50,52,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,32)(22,27,23,29)(25,34,33,31)(28,36,30,35)(37,38)(39,43)(40,47)(41,44)(42,45)(46,50)(48,51)(49,52);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,35)(29,36)(32,34)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,52)(47,51)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=g,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5521350);