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)(11,16)(12,17)(18,20)(24,31)(27,34)(29,35)(32,36)(38,42)(43,49)(44,47)(50,51);(7,13)(11,16)(14,19)(18,20)(21,22,25,28)(23,26,30,33)(24,27,31,34)(29,32,35,36)(37,38)(39,43)(40,44)(41,42)(45,50)(46,49)(47,48)(51,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,32)(30,33)(31,34)(35,36)(37,39,41,46)(38,43,42,49)(40,45,48,52)(44,50,47,51);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,30)(22,26,28,33)(24,29,31,35)(27,32,34,36)(37,38)(39,43)(40,47)(41,42)(44,48)(45,51)(46,49)(50,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,27)(23,29)(25,31)(26,32)(28,34)(30,35)(33,36)(37,40)(38,44)(39,45)(41,48)(42,47)(43,50)(46,52)(49,51)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=f,d^2=h,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, 4775863);