Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1540 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,28)(24,29)(25,30)(26,27)(31,36)(32,34)(33,35)(39,45)(41,49)(43,46)(44,48);(7,13)(10,17)(14,19)(18,20)(22,27)(23,29)(25,33)(32,36)(37,38,42,40)(39,46,45,43)(41,48,49,44)(47,50,51,52);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,23,26,29)(22,28,27,24)(25,34,33,31)(30,32,35,36)(37,39,42,45)(38,43,40,46)(41,47,49,51)(44,50,48,52);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,16)(19,20)(21,24,26,28)(22,29,27,23)(25,32,33,36)(30,34,35,31)(37,40,42,38)(39,43,45,46)(41,48,49,44)(47,50,51,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25)(22,30)(23,31)(24,32)(26,33)(27,35)(28,36)(29,34)(37,41)(38,44)(39,47)(40,48)(42,49)(43,50)(45,51)(46,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389140);