Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1642 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)(5,6,10,8)(7,13,14,11)(9,16,17,12)(15,18,20,19)(21,22,26,24)(23,27,29,31)(25,28,34,32)(30,35,33,36)(37,38,42,40)(39,43,47,45)(41,48,49,44)(46,51,52,50);(7,14)(9,17)(11,13)(12,16)(22,24)(23,29)(28,32)(30,33)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(5,7)(6,11)(8,13)(9,15)(10,14)(12,18)(16,19)(17,20)(21,23)(22,27)(24,31)(25,33)(26,29)(28,36)(30,34)(32,35)(37,39)(38,43)(40,45)(41,46)(42,47)(44,50)(48,51)(49,52);(1,2)(3,4)(5,8,10,6)(7,11,14,13)(9,16,17,12)(15,18,20,19)(21,24,26,22)(23,27,29,31)(25,32,34,28)(30,35,33,36)(37,40,42,38)(39,45,47,43)(41,48,49,44)(46,51,52,50);(5,9,10,17)(6,12,8,16)(7,15,14,20)(11,18,13,19)(21,25,26,34)(22,28,24,32)(23,30,29,33)(27,35,31,36)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5124605);