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 | #53188 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(9,10,15,16)(11,17,13,19)(12,22,23,18)(14,20,24,21)(25,26,32,36)(27,33,28,29)(30,34,45,51)(31,46,48,35)(37,43,39,41)(38,42,40,49)(44,52,47,50)(53,55,54,56);(1,2)(3,7)(4,6)(5,8)(10,16)(12,23)(17,19)(20,21)(25,27,32,28)(26,29,36,33)(30,39,45,37)(31,38,48,40)(34,43,51,41)(35,42,46,49)(44,54,47,53)(50,56,52,55);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,17,16,19)(12,21,23,20)(14,18,24,22)(25,28,32,27)(26,33,36,29)(30,39,45,37)(31,40,48,38)(34,43,51,41)(35,49,46,42)(44,54,47,53)(50,56,52,55);(2,6)(5,7)(9,12)(10,18)(11,21)(13,20)(14,19)(15,23)(16,22)(17,24)(25,29,32,33)(26,28,36,27)(30,43,45,41)(31,42,48,49)(34,37,51,39)(35,40,46,38)(44,56,47,55)(50,53,52,54);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,19,16,17)(12,21,23,20)(14,18,24,22)(25,30,32,45)(26,34,36,51)(27,37,28,39)(29,41,33,43)(31,47,48,44)(35,52,46,50)(38,54,40,53)(42,56,49,55);(1,3)(2,7)(4,8)(5,6)(9,14,15,24)(10,20,16,21)(11,18,13,22)(12,19,23,17)(25,31)(26,35)(27,38)(28,40)(29,42)(30,44)(32,48)(33,49)(34,50)(36,46)(37,53)(39,54)(41,55)(43,56)(45,47)(51,52)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10127588);