Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1566 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)(21,22,25,26)(23,29,31,27)(24,32,34,28)(30,35,33,36)(41,49)(44,48)(46,52)(50,51);(7,13)(11,16)(14,19)(18,20)(22,26)(27,29)(28,32)(35,36)(37,38,42,40)(39,45,47,43)(41,44,49,48)(46,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,22)(23,27)(24,28)(25,26)(29,31)(30,35)(32,34)(33,36)(37,40,42,38)(39,45,47,43)(41,48,49,44)(46,51,52,50);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=h,d^2=fgh,e^2=fg,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, 4517781);