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 | #2668 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4067 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,3)(4,7,8,6)(11,17)(14,20)(18,23)(21,24)(26,31)(27,35)(28,38)(34,41)(37,49)(40,52)(45,47)(55,56);(2,3)(6,7)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(26,31)(28,38)(32,43)(34,41)(36,39)(40,52)(46,51)(50,53);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,26)(27,32)(28,33)(29,41)(30,31)(34,42)(35,43)(36,45)(37,51)(38,44)(39,47)(40,54)(46,49)(48,52)(50,56)(53,55);(1,4)(2,6)(3,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,27,30,35)(26,32,31,43)(28,39,38,36)(29,37,42,49)(33,47,44,45)(34,46,41,51)(40,53,52,50)(48,56,54,55);(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,28,30,38)(26,33,31,44)(27,36,35,39)(29,40,42,52)(32,45,43,47)(34,48,41,54)(37,50,49,53)(46,55,51,56);(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,30,42)(26,34,31,41)(27,37,35,49)(28,40,38,52)(32,46,43,51)(33,48,44,54)(36,50,39,53)(45,55,47,56)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=g,d^2=gi,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, 8324451);