Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(10,14)(12,20)(15,18)(19,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40)(41,42)(43,45)(44,51)(46,54)(47,49)(48,55)(50,56)(52,53);(3,8)(5,7)(10,14)(11,17)(16,21)(19,23)(26,30)(27,34)(28,36)(38,39)(41,43,47,44)(42,45,49,51)(46,50,55,52)(48,53,54,56);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40)(41,44,47,43)(42,45,49,51)(46,52,55,50)(48,53,54,56);(1,3)(2,5)(4,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39)(41,45)(42,44)(43,49)(46,53)(47,51)(48,52)(50,54)(55,56);(1,2)(3,5)(4,6)(7,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,28,29,36)(26,32,30,37)(27,35,34,40)(31,38,33,39)(41,46,47,55)(42,48,49,54)(43,50,44,52)(45,53,51,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=fgh,c^2=fgh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5909049);