Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1558 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)(12,17)(14,19)(21,22)(23,29)(24,26)(25,28)(27,31)(30,35)(32,33)(34,36)(37,38)(39,42)(40,47)(41,44)(43,49)(45,52)(46,51)(48,50);(7,13)(11,16)(14,19)(18,20)(22,24)(27,29)(28,32)(34,35)(37,38,41,44)(39,42,46,51)(40,43,49,47)(45,50,48,52);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36)(37,38)(39,42)(40,43)(41,44)(45,50)(46,51)(47,49)(48,52);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35)(37,39,41,46)(38,42,44,51)(40,48,49,45)(43,52,47,50);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35)(37,40)(38,43)(39,45)(41,49)(42,50)(44,47)(46,48)(51,52)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=fgh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775484);