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 | |
1238 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)(9,10,14,12)(11,15,17,18)(13,20,21,16)(19,24,23,22)(25,26,30,31)(27,28,33,36)(29,37,39,32)(34,38,40,35)(42,46)(43,49)(44,52)(55,56);(11,17)(15,18)(19,23)(22,24)(26,31)(27,33)(32,37)(34,40)(41,42,45,46)(43,50,49,47)(44,48,52,54)(51,56,53,55);(9,11,14,17)(10,15,12,18)(13,19,21,23)(16,22,20,24)(25,27)(26,28)(29,34)(30,33)(31,36)(32,35)(37,38)(39,40)(41,42)(43,47)(44,48)(45,46)(49,50)(51,55)(52,54)(53,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,18,17,15)(13,20,21,16)(19,24,23,22)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39)(41,43,45,49)(42,47,46,50)(44,53,52,51)(48,56,54,55);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38)(41,44)(42,48)(43,51)(45,52)(46,54)(47,55)(49,53)(50,56)Polycyclic presentation:
2^9|a^2=gh,b^2=g,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, 4819488);