Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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)(21,22,26,30)(23,24,28,27)(25,33,34,29)(31,35,36,32)(38,42)(39,45)(44,48)(47,51);(6,10)(11,16)(12,17)(18,20)(23,28)(24,27)(31,36)(32,35)(37,38,41,42)(39,46,45,43)(40,44,49,48)(47,52,51,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,28)(22,27,30,24)(25,31,34,36)(29,35,33,32)(37,38)(39,43)(40,48)(41,42)(44,49)(45,46)(47,52)(50,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,27)(22,28,30,23)(25,32,34,35)(29,36,33,31)(37,39,41,45)(38,43,42,46)(40,47,49,51)(44,50,48,52);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25,26,34)(22,29,30,33)(23,31,28,36)(24,32,27,35)(37,40)(38,44)(39,47)(41,49)(42,48)(43,50)(45,51)(46,52)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fg,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4507667);