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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1556 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)(5,6,9,10)(7,13,14,11)(8,12,16,18)(15,20,17,19)(21,22)(23,24)(25,33)(26,27)(28,31)(29,35)(30,36)(32,34)(38,42)(43,45)(44,50)(51,52);(6,10)(7,14)(8,16)(19,20)(22,27)(24,31)(29,33)(32,36)(37,38,41,42)(39,45,47,43)(40,44,49,50)(46,52,48,51);(5,7,9,14)(6,11,10,13)(8,17,16,15)(12,20,18,19)(21,23)(22,24)(25,34)(26,28)(27,31)(29,36)(30,35)(32,33)(37,39,41,47)(38,43,42,45)(40,48,49,46)(44,52,50,51);(1,2,3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,24)(22,28)(23,27)(25,32)(26,31)(29,34)(30,33)(35,36)(37,38)(39,45)(40,44)(41,42)(43,47)(46,52)(48,51)(49,50);(5,8)(6,12)(7,15)(9,16)(10,18)(11,19)(13,20)(14,17)(21,25,26,35)(22,29,27,33)(23,30,28,34)(24,32,31,36)(37,40,41,49)(38,44,42,50)(39,46,47,48)(43,51,45,52)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,d^2=fi,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, 5188051);