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)(6,10)(7,13)(8,16)(18,20)(21,22)(23,28)(24,30)(25,29)(26,27)(31,33)(32,36)(34,35)(37,38,42,45)(39,46,40,43)(41,49,50,44)(47,51,48,52);(6,10)(8,16)(11,14)(15,19)(22,27)(23,30)(29,35)(31,36)(37,39,42,40)(38,43,45,46)(41,47,50,48)(44,51,49,52);(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23,26,30)(22,28,27,24)(25,31,34,36)(29,33,35,32)(37,40,42,39)(38,43,45,46)(41,48,50,47)(44,51,49,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,24,26,28)(22,23,27,30)(25,33,34,32)(29,36,35,31)(37,38)(39,43)(40,46)(41,49)(42,45)(44,50)(47,52)(48,51);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,29,27,35)(23,31,30,36)(24,32,28,33)(37,41,42,50)(38,44,45,49)(39,47,40,48)(43,51,46,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgh,c^2=f,d^2=h,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, 5524826);