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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,3)(4,7,8,6)(10,14)(12,20)(15,17)(18,24)(25,26,30,31)(27,28,32,36)(29,33,38,39)(34,35,40,37)(43,49)(44,51)(46,52)(47,54);(10,14)(12,20)(15,17)(18,24)(26,31)(27,32)(33,39)(34,40)(41,42,45,48)(43,46,49,52)(44,47,51,54)(50,53,55,56);(1,3,5,2)(4,7,8,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40)(41,42)(43,46)(44,47)(45,48)(49,52)(50,53)(51,54)(55,56);(1,4)(2,6)(3,7)(5,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,32)(27,31)(29,37)(30,36)(33,34)(35,38)(39,40)(41,43,45,49)(42,46,48,52)(44,50,51,55)(47,53,54,56);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29)(26,33)(27,34)(28,35)(30,38)(31,39)(32,40)(36,37)(41,44)(42,47)(43,50)(45,51)(46,53)(48,54)(49,55)(52,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgh,c^2=fghi,d^2=fh,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, 5638289);