Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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)(7,11)(8,10)(9,16)(12,18)(13,20)(14,15)(17,19)(21,22,26,27)(23,28,31,36)(24,25,30,29)(32,33,34,35)(37,38,43,46)(39,44,40,41)(42,45,50,52)(47,51,48,49);(6,8)(11,15)(12,16)(19,20)(22,27)(23,31)(24,30)(33,35)(37,39,43,40)(38,44,46,41)(42,48,50,47)(45,49,52,51);(5,7)(6,11)(8,15)(9,17)(10,14)(12,20)(13,18)(16,19)(21,23)(22,28)(24,34)(25,35)(26,31)(27,36)(29,33)(30,32)(37,40,43,39)(38,44,46,41)(42,48,50,47)(45,51,52,49);(1,2,3,4)(5,8,10,6)(7,11,14,15)(9,16,18,12)(13,19,17,20)(21,24,26,30)(22,29,27,25)(23,32,31,34)(28,35,36,33)(37,41,43,44)(38,40,46,39)(42,49,50,51)(45,48,52,47);(5,9)(6,12)(7,13)(8,16)(10,18)(11,19)(14,17)(15,20)(21,25,26,29)(22,30,27,24)(23,33,31,35)(28,34,36,32)(37,42)(38,45)(39,47)(40,48)(41,49)(43,50)(44,51)(46,52)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5189691);