Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,8)(9,10,13,14)(11,17,18,15)(12,20,22,16)(19,23,21,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38)(41,42,46,49)(43,44,51,47)(45,53,55,48)(50,54,56,52);(10,14)(11,18)(16,20)(19,21)(27,33)(31,34)(35,39)(38,40)(41,43,46,51)(42,47,49,44)(45,50,55,56)(48,54,53,52);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40)(41,43)(42,44)(45,50)(46,51)(47,49)(48,52)(53,54)(55,56);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,15,14,17)(12,21,22,19)(16,24,20,23)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38)(41,44)(42,43)(45,54)(46,47)(48,56)(49,51)(50,53)(52,55);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)(41,45,46,55)(42,48,49,53)(43,50,51,56)(44,52,47,54)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fh,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4819458);