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)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33)(37,38,42,40)(39,43,47,45)(41,48,49,44)(46,51,52,50);(6,7)(11,13)(12,14)(19,20)(24,32)(27,34)(30,36)(33,35)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(5,7,10,6)(8,11,16,13)(9,14,18,12)(15,19,17,20)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35)(37,39)(38,43)(40,45)(41,46)(42,47)(44,50)(48,51)(49,52);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,17,18,15)(12,20,14,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36)(37,40,42,38)(39,45,47,43)(41,48,49,44)(46,51,52,50);(5,9,10,18)(6,12,7,14)(8,15,16,17)(11,19,13,20)(21,24)(22,27)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=gh,b^2=fh,c^2=fg,d^2=h,e^2=f,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, 5524377);