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 | #52403 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(2,5)(6,8)(10,14)(11,17)(12,20)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,46)(36,38)(37,47)(39,48)(40,42)(41,49)(43,50)(44,45)(51,55)(52,56)(53,54);(3,7)(6,8)(9,10,15,14)(11,18,17,13)(12,16,20,22)(19,24,23,21)(26,31)(32,36)(33,40)(34,45)(46,48)(47,50)(49,53)(55,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,19)(14,18)(15,17)(16,21)(20,23)(22,24)(25,27)(26,32)(28,39)(29,43)(30,38)(31,36)(33,48)(34,50)(35,42)(37,44)(40,46)(41,51)(45,47)(49,55)(52,54)(53,56);(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24)(25,28)(26,33)(27,35)(29,41)(30,42)(31,40)(32,46)(34,49)(36,48)(37,51)(38,39)(43,52)(44,54)(45,53)(47,55)(50,56);(1,3)(2,6)(4,7)(5,8)(9,13,15,18)(10,17,14,11)(12,21,20,24)(16,23,22,19)(25,26)(27,36)(28,40)(29,34)(30,31)(32,38)(33,42)(35,46)(37,50)(39,48)(41,53)(43,47)(44,45)(49,54)(51,55)(52,56);(1,2)(3,6)(4,5)(7,8)(9,14,15,10)(11,18,17,13)(12,22,20,16)(19,24,23,21)(25,29,30,44)(26,34,31,45)(27,37,38,43)(28,41,42,54)(32,47,36,50)(33,49,40,53)(35,51,39,52)(46,55,48,56)Polycyclic presentation:
2^9|b^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8961052);