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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,8)(3,6,4,7)(10,11)(13,21)(15,17)(19,24)(25,26)(27,29)(28,33)(30,43)(31,32)(34,44)(35,42)(36,39)(37,50)(38,45)(40,54)(41,48)(46,52)(47,51)(49,56)(53,55);(1,2)(3,6)(4,7)(5,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(26,32)(27,35)(28,38)(34,43)(37,48)(39,51)(40,52)(53,56);(1,3,5,4)(2,6,8,7)(9,11,14,10)(12,15,20,17)(13,18,21,16)(19,22,24,23)(25,27,31,35)(26,29,32,42)(28,36,38,47)(30,37,44,48)(33,39,45,51)(34,41,43,50)(40,49,52,55)(46,53,54,56);(1,4,5,3)(2,7,8,6)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,31,38)(26,33,32,45)(27,36,35,47)(29,39,42,51)(30,40,44,52)(34,46,43,54)(37,49,48,55)(41,53,50,56);(9,13)(10,16)(11,18)(12,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,27)(28,39)(30,41)(31,42)(32,35)(33,36)(34,37)(38,51)(40,53)(43,48)(44,50)(45,47)(46,49)(52,56)(54,55);(25,30,31,44)(26,34,32,43)(27,37,35,48)(28,40,38,52)(29,41,42,50)(33,46,45,54)(36,49,47,55)(39,53,51,56)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622640);