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)(7,10)(8,13)(9,12)(11,15)(14,17)(16,19)(18,20)(21,22,25,26)(23,27,31,29)(24,32,34,28)(30,36,33,35)(37,38,43,46)(39,44,40,41)(42,45,50,52)(47,51,48,49);(6,7)(8,15)(9,17)(18,19)(22,26)(27,29)(28,32)(35,36)(37,39,43,40)(38,44,46,41)(42,48,50,47)(45,49,52,51);(5,7,10,6)(8,13,15,11)(9,12,17,14)(16,18,20,19)(21,23,25,31)(22,27,26,29)(24,33,34,30)(28,36,32,35)(37,40,43,39)(38,44,46,41)(42,48,50,47)(45,51,52,49);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,34)(33,35)(37,41,43,44)(38,40,46,39)(42,49,50,51)(45,48,52,47);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,24,25,34)(22,28,26,32)(23,30,31,33)(27,35,29,36)(37,42)(38,45)(39,47)(40,48)(41,49)(43,50)(44,51)(46,52)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fgh,d^2=fi,e^2=f,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, 5187672);