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 | |
3983 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)(4,8)(6,7)(11,13)(15,17)(19,21)(23,24)(25,26,31,35)(27,29,33,42)(28,38,41,32)(30,34,46,50)(36,39,48,43)(37,44,45,49)(40,52,55,47)(51,53,54,56);(2,7)(3,8)(9,10,14,18)(11,15,13,17)(12,20,22,16)(19,24,21,23)(27,33)(29,42)(36,48)(37,45)(39,43)(44,49)(51,54)(53,56);(1,3,6,8)(2,4,7,5)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,27)(26,29)(28,36)(30,45)(31,33)(32,43)(34,49)(35,42)(37,46)(38,39)(40,54)(41,48)(44,50)(47,53)(51,55)(52,56);(1,4,6,5)(2,3,7,8)(25,28,31,41)(26,32,35,38)(27,36,33,48)(29,43,42,39)(30,40,46,55)(34,47,50,52)(37,51,45,54)(44,56,49,53);(1,2)(3,4)(5,8)(6,7)(9,12,14,22)(10,16,18,20)(11,19,13,21)(15,23,17,24)(25,29)(26,33)(27,35)(28,39)(30,44)(31,42)(32,36)(34,45)(37,50)(38,48)(40,53)(41,43)(46,49)(47,51)(52,54)(55,56);(1,5,6,4)(2,8,7,3)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(25,30)(26,34)(27,37)(28,40)(29,44)(31,46)(32,47)(33,45)(35,50)(36,51)(38,52)(39,53)(41,55)(42,49)(43,56)(48,54)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=i,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946360);