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 | #53053 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4093 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)(10,14)(12,19)(15,20)(18,23)(25,26,31,35)(27,29,36,32)(28,33,39,48)(30,44,45,34)(37,41,40,46)(38,47,50,43)(42,55,52,49)(51,56,53,54);(2,5)(6,7)(11,17)(15,20)(18,23)(22,24)(27,36)(28,39)(29,32)(33,48)(38,50)(42,52)(43,47)(49,55);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27)(26,32)(28,40)(29,35)(30,38)(31,36)(33,41)(34,47)(37,39)(42,53)(43,44)(45,50)(46,48)(49,54)(51,52)(55,56);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28)(26,33)(27,37)(29,41)(30,42)(31,39)(32,46)(34,49)(35,48)(36,40)(38,51)(43,54)(44,55)(45,52)(47,56)(50,53);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,31,32)(26,27,35,36)(28,41,39,46)(30,43,45,47)(33,37,48,40)(34,38,44,50)(42,54,52,56)(49,51,55,53);(25,30,31,45)(26,34,35,44)(27,38,36,50)(28,42,39,52)(29,43,32,47)(33,49,48,55)(37,51,40,53)(41,54,46,56)Polycyclic presentation:
2^9|a^2=gi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8022633);