Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,4)(6,7)(9,10)(11,17)(12,14)(13,20)(15,18)(16,21)(19,22)(23,24)(25,26,30,31)(27,35,38,32)(28,39,40,33)(29,43,44,34)(36,45,41,47)(37,46,51,49)(42,48,53,52)(50,56,54,55);(1,2,5,4)(3,7,8,6)(10,12)(11,18)(16,20)(19,24)(26,31)(28,40)(32,35)(34,43)(36,41)(42,53)(46,49)(50,54);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27)(26,32)(28,41)(29,37)(30,38)(31,35)(33,47)(34,46)(36,40)(39,45)(42,54)(43,49)(44,51)(48,56)(50,53)(52,55);(1,4,5,2)(3,7,8,6)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,28,30,40)(26,33,31,39)(27,36,38,41)(29,42,44,53)(32,45,35,47)(34,48,43,52)(37,50,51,54)(46,55,49,56);(9,13)(10,16)(11,19)(12,20)(14,21)(15,22)(17,23)(18,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,45)(37,46)(39,40)(41,47)(42,48)(43,44)(49,51)(50,55)(52,53)(54,56);(25,29,30,44)(26,34,31,43)(27,37,38,51)(28,42,40,53)(32,46,35,49)(33,48,39,52)(36,50,41,54)(45,55,47,56)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022569);