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)(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(21,22,26,27)(23,30,28,24)(25,29,33,35)(31,34,36,32)(38,40)(43,45)(44,48)(50,51);(7,13)(11,14)(15,19)(18,20)(22,27)(23,28)(25,33)(32,34)(37,38,42,40)(39,45,47,43)(41,44,49,48)(46,51,52,50);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23,26,28)(22,24,27,30)(25,31,33,36)(29,32,35,34)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2)(3,4)(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24)(22,28)(23,27)(25,34)(26,30)(29,31)(32,33)(35,36)(37,40,42,38)(39,43,47,45)(41,44,49,48)(46,51,52,50);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25)(22,29)(23,31)(24,32)(26,33)(27,35)(28,36)(30,34)(37,41,42,49)(38,44,40,48)(39,46,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=h,d^2=gh,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, 5527126);