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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,13)(8,12)(9,10)(11,14)(15,19)(16,17)(18,20)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(37,38)(39,44)(40,46)(41,43)(42,45)(47,51)(48,52)(49,50);(6,10)(7,14)(12,16)(15,20)(23,29)(26,33)(30,35)(34,36)(37,39,43,40)(38,44,41,46)(42,48,50,47)(45,52,49,51);(5,6)(7,13)(8,16)(9,10)(11,14)(12,17)(15,18)(19,20)(21,23)(22,26)(24,30)(25,29)(27,34)(28,33)(31,36)(32,35)(37,40,43,39)(38,44,41,46)(42,48,50,47)(45,51,49,52);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,18)(16,19)(17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,41,43,38)(39,46,40,44)(42,49,50,45)(47,52,48,51);(5,8)(6,12)(7,15)(9,17)(10,16)(11,18)(13,19)(14,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,42,43,50)(38,45,41,49)(39,47,40,48)(44,51,46,52)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=g,d^2=hi,e^2=fgh,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, 5192236);