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 | |
1556 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)(5,6,9,10)(7,11,15,13)(8,16,18,12)(14,20,17,19)(21,22)(23,25)(24,31)(26,27)(28,32)(29,35)(30,36)(33,34)(37,38,43,46)(39,44,40,41)(42,45,50,52)(47,51,48,49);(6,10)(11,13)(12,16)(19,20)(22,27)(23,29)(24,32)(34,36)(37,39,43,40)(38,44,46,41)(42,48,50,47)(45,49,52,51);(5,7,9,15)(6,11,10,13)(8,17,18,14)(12,20,16,19)(21,23,26,29)(22,25,27,35)(24,33,32,30)(28,36,31,34)(37,40,43,39)(38,44,46,41)(42,48,50,47)(45,51,52,49);(1,2,3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,18)(17,19)(21,24)(22,28)(23,30)(25,34)(26,32)(27,31)(29,33)(35,36)(37,41,43,44)(38,40,46,39)(42,49,50,51)(45,48,52,47);(5,8,9,18)(6,12,10,16)(7,14,15,17)(11,19,13,20)(21,25)(22,23)(24,34)(26,35)(27,29)(28,30)(31,33)(32,36)(37,42)(38,45)(39,47)(40,48)(41,49)(43,50)(44,51)(46,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=fi,e^2=fh,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, 5189720);