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 | #392 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3987 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)(3,5)(4,7)(6,8)(11,17)(12,19)(14,20)(15,22)(26,31)(27,35)(29,42)(33,44)(37,39)(40,53)(46,50)(55,56);(1,2,6,8)(3,5,7,4)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(26,31)(32,36)(33,44)(34,43)(45,47)(46,50)(48,54)(55,56);(1,3,6,7)(2,4,8,5)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,26)(27,36)(28,33)(29,43)(30,31)(32,35)(34,42)(37,47)(38,46)(39,45)(40,54)(41,44)(48,53)(49,50)(51,55)(52,56);(1,4,6,5)(2,3,8,7)(25,27)(26,32)(28,39)(29,38)(30,35)(31,36)(33,47)(34,46)(37,41)(40,52)(42,49)(43,50)(44,45)(48,56)(51,53)(54,55);(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,22)(23,24)(25,28,30,41)(26,33,31,44)(27,37,35,39)(29,40,42,53)(32,45,36,47)(34,48,43,54)(38,51,49,52)(46,55,50,56);(1,5,6,4)(2,7,8,3)(9,12,13,19)(10,15,16,22)(11,18,17,23)(14,21,20,24)(25,29,30,42)(26,34,31,43)(27,38,35,49)(28,40,41,53)(32,46,36,50)(33,48,44,54)(37,51,39,52)(45,55,47,56)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839357);