Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1270 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,4,7,8)(10,14)(15,17)(16,21)(22,23)(25,26,30,27)(28,31,35,33)(29,34,37,32)(36,39,40,38)(44,51)(45,53)(47,49)(48,50);(3,7)(4,8)(12,20)(16,21)(18,24)(22,23)(26,27)(28,35)(32,34)(36,40)(41,42,46,43)(44,49,51,47)(45,48,53,50)(52,55,56,54);(1,3)(2,4)(5,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,27,30,26)(28,31,35,33)(29,32,37,34)(36,39,40,38)(41,43,46,42)(44,49,51,47)(45,50,53,48)(52,55,56,54);(1,4)(2,3)(5,8)(6,7)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40)(41,44)(42,47)(43,49)(45,52)(46,51)(48,54)(50,55)(53,56);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)(41,45)(42,48)(43,50)(44,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=hi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6178180);