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 | |
1554 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)(7,13)(8,15)(11,16)(12,17)(21,22,26,27)(23,28,29,24)(25,30,34,35)(31,33,36,32)(37,38)(39,44)(40,46)(41,43)(42,45)(47,51)(48,52)(49,50);(6,10)(11,16)(12,17)(18,20)(22,27)(23,29)(30,35)(31,36)(37,39,43,40)(38,44,41,46)(42,48,50,47)(45,52,49,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,33)(32,35)(34,36)(37,40,43,39)(38,44,41,46)(42,48,50,47)(45,51,49,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36)(37,41,43,38)(39,46,40,44)(42,49,50,45)(47,52,48,51);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,30)(23,31)(24,32)(26,34)(27,35)(28,33)(29,36)(37,42,43,50)(38,45,41,49)(39,47,40,48)(44,51,46,52)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fgh,d^2=gh,e^2=g,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, 5526184);