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)(7,10)(8,13)(9,12)(11,16)(14,17)(15,19)(18,20)(21,22,25,28)(23,29,31,26)(24,27,32,34)(30,35,36,33)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(6,7)(11,13)(12,14)(18,19)(24,32)(27,34)(30,36)(33,35)(37,38,41,44)(39,42,47,45)(40,48,49,43)(46,51,52,50);(5,7,10,6)(8,11,16,13)(9,12,17,14)(15,19,20,18)(21,23,25,31)(22,26,28,29)(24,30,32,36)(27,33,34,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,22)(23,29)(24,27)(25,28)(26,31)(30,35)(32,34)(33,36)(37,39)(38,42)(40,46)(41,47)(43,50)(44,45)(48,51)(49,52);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,32)(22,27,28,34)(23,30,31,36)(26,33,29,35)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fgh,d^2=f,e^2=fg,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, 5383034);