Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1625 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)(21,22,26,27)(23,24,28,32)(25,29,34,35)(30,31,36,33)(38,42)(39,45)(44,47)(46,48);(7,13)(11,14)(15,19)(18,20)(22,27)(23,28)(29,35)(30,36)(37,38,41,42)(39,43,45,50)(40,47,49,44)(46,52,48,51);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,35)(34,36)(37,38)(39,43)(40,44)(41,42)(45,50)(46,51)(47,49)(48,52);(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)(25,33)(26,32)(29,30)(31,34)(35,36)(37,39,41,45)(38,43,42,50)(40,48,49,46)(44,52,47,51);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25)(22,29)(23,30)(24,31)(26,34)(27,35)(28,36)(32,33)(37,40,41,49)(38,44,42,47)(39,46,45,48)(43,51,50,52)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=fh,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5527119);