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)(12,17)(15,19)(21,22,26,27)(23,30,28,24)(25,29,33,35)(31,34,36,32)(37,38,42,45)(39,46,40,43)(41,49,50,44)(47,51,48,52);(8,16)(12,17)(15,19)(18,20)(22,27)(23,28)(25,33)(32,34)(37,39,42,40)(38,43,45,46)(41,47,50,48)(44,51,49,52);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23,26,28)(22,24,27,30)(25,31,33,36)(29,32,35,34)(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)(22,28)(23,27)(25,34)(26,30)(29,31)(32,33)(35,36)(37,38)(39,43)(40,46)(41,49)(42,45)(44,50)(47,52)(48,51);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25)(22,29)(23,31)(24,32)(26,33)(27,35)(28,36)(30,34)(37,41,42,50)(38,44,45,49)(39,47,40,48)(43,51,46,52)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=gh,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, 5544058);