Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(7,13)(8,15)(18,20)(21,22,26,23)(24,29,32,27)(25,28,33,30)(31,35,36,34)(41,49)(44,48)(46,52)(50,51);(6,10)(7,13)(12,17)(14,19)(22,23)(27,29)(28,30)(34,35)(37,38,42,40)(39,45,47,43)(41,44,49,48)(46,51,52,50);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23,26,22)(24,29,32,27)(25,28,33,30)(31,34,36,35)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36)(37,40,42,38)(39,45,47,43)(41,48,49,44)(46,51,52,50);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25,26,33)(22,28,23,30)(24,31,32,36)(27,34,29,35)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=h,d^2=fgh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4517519);