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, 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)(5,6,10,7)(8,13,16,11)(9,12,18,14)(15,20,17,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38)(39,42)(40,48)(41,45)(43,47)(44,50)(46,52)(49,51);(6,7)(11,13)(12,14)(19,20)(23,29)(27,30)(31,35)(34,36)(37,39,41,47)(38,42,45,43)(40,49,50,46)(44,52,48,51);(5,7,10,6)(8,11,16,13)(9,14,18,12)(15,19,17,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35)(37,39)(38,43)(40,46)(41,47)(42,45)(44,52)(48,51)(49,50);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,17,18,15)(12,20,14,19)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,38)(39,42)(40,44)(41,45)(43,47)(46,51)(48,50)(49,52);(5,9,10,18)(6,12,7,14)(8,15,16,17)(11,19,13,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)(37,40)(38,44)(39,46)(41,50)(42,51)(43,52)(45,48)(47,49)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=gh,e^2=gh,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, 5539399);