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)(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,36)(34,35)(38,42)(39,45)(44,48)(47,51);(7,13)(11,14)(15,19)(18,20)(22,27)(23,28)(29,35)(30,33)(37,38,41,42)(39,46,45,43)(40,44,49,48)(47,52,51,50);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23,26,28)(22,24,27,32)(25,33,34,30)(29,36,35,31)(37,38)(39,43)(40,48)(41,42)(44,49)(45,46)(47,52)(50,51);(1,2)(3,4)(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24)(22,28)(23,27)(25,31)(26,32)(29,33)(30,35)(34,36)(37,39,41,45)(38,43,42,46)(40,47,49,51)(44,50,48,52);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,29,27,35)(23,30,28,33)(24,31,32,36)(37,40)(38,44)(39,47)(41,49)(42,48)(43,50)(45,51)(46,52)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=fg,d^2=fgh,e^2=gh,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, 4513449);