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 | |
4011 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)(9,10)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(26,31)(28,39)(29,42)(32,36)(37,47)(38,48)(46,53)(54,56);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(26,31)(27,35)(33,40)(34,43)(37,47)(38,48)(46,53)(51,55);(1,3,5,4)(2,6,8,7)(10,16)(15,18)(17,20)(23,24)(25,26)(27,36)(28,40)(29,34)(30,31)(32,35)(33,39)(37,44)(38,50)(41,53)(42,43)(45,48)(46,52)(47,49)(51,54)(55,56);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,27,30,35)(26,32,31,36)(28,37,39,47)(29,38,42,48)(33,44,40,49)(34,45,43,50)(41,51,52,55)(46,54,53,56);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,28)(26,33)(27,37)(29,41)(30,39)(31,40)(32,44)(34,46)(35,47)(36,49)(38,51)(42,52)(43,53)(45,54)(48,55)(50,56);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,29,30,42)(26,34,31,43)(27,38,35,48)(28,41,39,52)(32,45,36,50)(33,46,40,53)(37,51,47,55)(44,54,49,56)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=g,d^2=gi,e^2=g,f^2=h,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, 7621655);