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 | #56916 | C2 |
| 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) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(27,35)(31,36)(37,40)(38,49)(45,47)(46,50)(51,53)(55,56);(2,5)(6,7)(10,16)(11,12)(18,23)(19,20)(25,26,30,34)(27,36,35,31)(28,39,42,32)(29,33,44,43)(37,45,40,47)(38,50,49,46)(41,52,54,48)(51,55,53,56);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,26)(27,31)(28,32)(29,43)(30,34)(33,44)(35,36)(37,45)(38,50)(39,42)(40,47)(41,52)(46,49)(48,54)(51,56)(53,55);(1,3)(2,6)(4,8)(5,7)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,27,30,35)(26,31,34,36)(28,40,42,37)(29,38,44,49)(32,47,39,45)(33,46,43,50)(41,53,54,51)(48,56,52,55);(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,28,30,42)(26,32,34,39)(27,37,35,40)(29,41,44,54)(31,45,36,47)(33,48,43,52)(38,51,49,53)(46,55,50,56);(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,29)(26,33)(27,38)(28,41)(30,44)(31,46)(32,48)(34,43)(35,49)(36,50)(37,51)(39,52)(40,53)(42,54)(45,55)(47,56)Polycyclic presentation:
2^9|a^2=h,b^2=gi,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, 8324493);