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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(10,15)(11,17)(16,23)(18,20)(25,26,31,28)(27,35,38,32)(29,33,41,39)(30,40,43,34)(36,47,49,44)(37,45,51,48)(42,52,53,46)(50,54,56,55);(1,2)(3,4)(5,7)(6,8)(9,10,14,15)(11,13,17,22)(12,16,19,23)(18,21,20,24)(29,41)(33,39)(36,49)(42,53)(44,47)(46,52)(50,56)(54,55);(1,3,5,8)(2,6,7,4)(12,19)(16,23)(18,20)(21,24)(25,27,31,38)(26,32,28,35)(29,36,41,49)(30,37,43,51)(33,44,39,47)(34,45,40,48)(42,50,53,56)(46,54,52,55);(1,3,5,8)(2,4,7,6)(9,11)(10,13)(12,20)(14,17)(15,22)(16,24)(18,19)(21,23)(25,28,31,26)(27,32,38,35)(29,39,41,33)(30,40,43,34)(36,44,49,47)(37,45,51,48)(42,52,53,46)(50,54,56,55);(1,3)(2,4)(5,8)(6,7)(9,12,14,19)(10,16,15,23)(11,18,17,20)(13,21,22,24)(25,29)(26,33)(27,36)(28,39)(30,42)(31,41)(32,44)(34,46)(35,47)(37,50)(38,49)(40,52)(43,53)(45,54)(48,55)(51,56);(1,4)(2,3)(5,6)(7,8)(9,13)(10,17)(11,15)(12,21)(14,22)(16,20)(18,23)(19,24)(25,30)(26,34)(27,37)(28,40)(29,42)(31,43)(32,45)(33,46)(35,48)(36,50)(38,51)(39,52)(41,53)(44,54)(47,55)(49,56)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=hi,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,e^d=egi,f^d=fgi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9542145);