Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1609 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)(7,13)(8,15)(11,16)(12,17)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35)(38,42)(39,45)(44,48)(47,51);(6,10)(11,16)(12,17)(18,20)(22,23)(25,33)(27,29)(31,36)(37,38,41,42)(39,43,45,46)(40,48,49,44)(47,52,51,50);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35)(37,39)(38,43)(40,47)(41,45)(42,46)(44,50)(48,52)(49,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35)(37,38)(39,46)(40,44)(41,42)(43,45)(47,52)(48,49)(50,51);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)(37,40,41,49)(38,44,42,48)(39,47,45,51)(43,50,46,52)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=h,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4376605);