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 | |
1570 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)(6,10)(7,13)(12,17)(15,19)(21,22,26,27)(23,28,29,24)(25,30,34,35)(31,33,36,32)(39,45)(40,47)(42,48)(43,50);(8,16)(12,17)(15,19)(18,20)(22,27)(23,29)(30,35)(31,36)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23)(22,28)(24,27)(25,31)(26,29)(30,33)(32,35)(34,36)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36)(37,39,41,45)(38,42,44,48)(40,46,47,51)(43,49,50,52);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,30,27,35)(23,31,29,36)(24,32,28,33)(37,40)(38,43)(39,46)(41,47)(42,49)(44,50)(45,51)(48,52)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=g,d^2=gh,e^2=fg,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, 5524086);