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 | #2678 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26)(27,35)(28,33)(29,31)(30,34)(32,38)(36,49)(37,48)(39,47)(40,41)(42,44)(43,46)(45,51)(50,56)(52,55)(53,54);(1,2,4,6)(3,5,8,7)(10,16)(11,12)(18,23)(19,20)(26,29)(32,35)(33,41)(34,42)(45,49)(46,48)(47,54)(55,56);(1,2)(3,5)(4,6)(7,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27,31,38)(26,32,29,35)(28,36,40,51)(30,43,44,37)(33,45,41,49)(34,48,42,46)(39,52,53,50)(47,56,54,55);(1,3)(2,5)(4,8)(6,7)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28)(26,33)(27,36)(29,41)(30,39)(31,40)(32,45)(34,47)(35,49)(37,50)(38,51)(42,54)(43,52)(44,53)(46,55)(48,56);(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29,31,26)(27,35,38,32)(28,33,40,41)(30,42,44,34)(36,45,51,49)(37,48,43,46)(39,47,53,54)(50,55,52,56);(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30,31,44)(26,34,29,42)(27,37,38,43)(28,39,40,53)(32,46,35,48)(33,47,41,54)(36,50,51,52)(45,55,49,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=gh,c^2=h,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8324330);