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 | |
3992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34)(27,31,38,35)(28,32,39,45)(29,41,42,33)(36,43,49,47)(37,48,51,44)(40,53,52,46)(50,55,56,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,13,20,23)(16,22,24,17)(28,39)(32,45)(36,49)(40,52)(43,47)(46,53)(50,56)(54,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,16,18,24)(11,13,21,23)(15,22,19,17)(25,26)(27,35)(28,32)(29,33)(30,34)(31,38)(36,47)(37,48)(39,45)(40,46)(41,42)(43,49)(44,51)(50,55)(52,53)(54,56);(2,6)(5,7)(9,11)(10,15)(12,13)(14,21)(16,22)(17,24)(18,19)(20,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,43)(33,44)(34,35)(39,49)(40,50)(41,48)(42,51)(45,47)(46,54)(52,56)(53,55);(1,2)(3,5)(4,6)(7,8)(9,13)(10,17)(11,20)(12,21)(14,23)(15,16)(18,22)(19,24)(25,28)(26,32)(27,36)(29,40)(30,39)(31,43)(33,46)(34,45)(35,47)(37,50)(38,49)(41,53)(42,52)(44,54)(48,55)(51,56);(1,3)(2,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,22)(14,18)(17,23)(19,21)(20,24)(25,29,30,42)(26,33,34,41)(27,37,38,51)(28,40,39,52)(31,44,35,48)(32,46,45,53)(36,50,49,56)(43,54,47,55)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10126983);