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 | |
1626 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)(5,6,9,10)(7,13,15,11)(8,16,17,12)(14,19,18,20)(21,22)(23,29)(24,26)(25,28)(27,30)(31,36)(32,33)(34,35)(37,38,42,45)(39,43,40,47)(41,44,49,52)(46,50,48,51);(6,10)(8,17)(11,13)(14,18)(22,24)(23,30)(25,33)(35,36)(37,39,42,40)(38,43,45,47)(41,48,49,46)(44,51,52,50);(5,7)(6,11)(8,18)(9,15)(10,13)(12,20)(14,17)(16,19)(21,23)(22,27)(24,29)(25,34)(26,30)(28,36)(31,33)(32,35)(37,40,42,39)(38,43,45,47)(41,48,49,46)(44,50,52,51);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,19)(16,17)(18,20)(21,24,26,22)(23,29,30,27)(25,32,33,28)(31,36,34,35)(37,38)(39,43)(40,47)(41,44)(42,45)(46,50)(48,51)(49,52);(5,8,9,17)(6,12,10,16)(7,14,15,18)(11,19,13,20)(21,25,26,33)(22,28,24,32)(23,31,30,34)(27,35,29,36)(37,41,42,49)(38,44,45,52)(39,46,40,48)(43,50,47,51)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fg,d^2=h,e^2=g,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, 5123649);