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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1636 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)(21,22,26,27)(23,24,28,32)(25,29,34,35)(30,31,36,33)(37,38,42,45)(39,43,40,47)(41,44,49,52)(46,50,48,51);(7,14)(9,17)(11,13)(12,16)(22,27)(23,28)(29,35)(30,36)(37,39,42,40)(38,43,45,47)(41,48,49,46)(44,51,52,50);(5,7)(6,11)(8,13)(9,15)(10,14)(12,18)(16,19)(17,20)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,35)(34,36)(37,40,42,39)(38,43,45,47)(41,48,49,46)(44,50,52,51);(1,2)(3,4)(5,8,10,6)(7,11,14,13)(9,16,17,12)(15,18,20,19)(21,24)(22,28)(23,27)(25,33)(26,32)(29,30)(31,34)(35,36)(37,38)(39,43)(40,47)(41,44)(42,45)(46,50)(48,51)(49,52);(5,9,10,17)(6,12,8,16)(7,15,14,20)(11,18,13,19)(21,25)(22,29)(23,30)(24,31)(26,34)(27,35)(28,36)(32,33)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,52)(47,51)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=fgh,d^2=fh,e^2=fh,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, 5543906);