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×#121 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,35)(28,39)(29,31)(30,43)(32,38)(33,40)(34,44)(36,45)(37,46)(41,47)(42,48)(49,51)(50,56)(52,53)(54,55);(2,5)(6,7)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(26,29)(28,40)(32,35)(34,43)(36,41)(42,53)(46,49)(50,54);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27)(26,32)(28,41)(29,35)(30,37)(31,38)(33,47)(34,46)(36,40)(39,45)(42,54)(43,49)(44,51)(48,56)(50,53)(52,55);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28)(26,33)(27,36)(29,39)(30,42)(31,40)(32,45)(34,48)(35,47)(37,50)(38,41)(43,52)(44,53)(46,55)(49,56)(51,54);(1,3,4,8)(2,6,5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,29,31,26)(27,35,38,32)(28,39,40,33)(30,43,44,34)(36,47,41,45)(37,49,51,46)(42,52,53,48)(50,56,54,55);(25,30,31,44)(26,34,29,43)(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=ghi,b^2=gh,d^2=gh,e^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, 8022610);