Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 4194304 |
| 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 | |
1606 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)(10,16)(11,18)(21,22,26,23)(24,27,32,29)(25,30,33,28)(31,35,36,34)(38,39)(43,45)(44,46)(50,51);(1,2)(3,4)(7,13)(10,16)(14,19)(17,20)(22,23)(27,29)(28,30)(34,35)(37,38,42,39)(40,45,48,43)(41,46,49,44)(47,50,52,51);(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,23,26,22)(24,29,32,27)(25,30,33,28)(31,35,36,34)(37,39,42,38)(40,45,48,43)(41,46,49,44)(47,51,52,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35)(37,40)(38,43)(39,45)(41,47)(42,48)(44,50)(46,51)(49,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)(37,41,42,49)(38,44,39,46)(40,47,48,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=ghi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039787);