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 | #57972 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4099 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)(4,6)(5,8)(9,10)(11,19)(12,23)(13,24)(14,18)(15,16)(17,20)(21,22)(25,26,31,35)(27,32,36,29)(28,39,40,33)(30,34,46,50)(37,43,48,41)(38,47,45,44)(42,52,53,49)(51,56,55,54);(1,2,4,6)(3,5,8,7)(10,16)(11,20)(12,24)(18,22)(27,36)(28,40)(29,32)(33,39)(38,45)(42,53)(44,47)(49,52);(1,2)(3,5)(4,6)(7,8)(9,11,15,20)(10,17,16,19)(12,21,24,14)(13,22,23,18)(25,27)(26,32)(28,37)(29,35)(30,45)(31,36)(33,41)(34,44)(38,46)(39,43)(40,48)(42,55)(47,50)(49,56)(51,53)(52,54);(1,3)(2,5)(4,8)(6,7)(9,12)(10,13)(11,21)(14,20)(15,24)(16,23)(17,22)(18,19)(25,28,31,40)(26,33,35,39)(27,37,36,48)(29,43,32,41)(30,42,46,53)(34,49,50,52)(38,51,45,55)(44,56,47,54);(9,13)(10,12)(11,22)(14,19)(15,23)(16,24)(17,21)(18,20)(25,29,31,32)(26,27,35,36)(28,41,40,43)(30,44,46,47)(33,48,39,37)(34,38,50,45)(42,54,53,56)(49,55,52,51);(9,14,15,21)(10,18,16,22)(11,12,20,24)(13,19,23,17)(25,30)(26,34)(27,38)(28,42)(29,44)(31,46)(32,47)(33,49)(35,50)(36,45)(37,51)(39,52)(40,53)(41,54)(43,56)(48,55)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,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, 8324286);