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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 56 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,19)(13,21)(14,17)(16,22)(18,24)(20,23)(25,26)(27,31)(28,32)(29,36)(30,37)(33,42)(34,43)(35,38)(39,47)(40,48)(41,46)(44,50)(45,51)(49,53)(52,54)(55,56);(1,3,5,8)(2,6,4,7)(9,11,14,12)(10,15,17,19)(13,18,22,20)(16,23,21,24)(26,27)(28,38)(33,36)(34,37)(39,50)(40,51)(46,49)(52,56);(9,12,14,11)(10,15,17,19)(13,20,22,18)(16,23,21,24)(25,27,31,26)(28,32,38,35)(29,33,42,36)(30,37,43,34)(39,47,50,44)(40,45,51,48)(41,46,53,49)(52,55,56,54);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,19)(13,16)(14,17)(18,23)(20,24)(21,22)(25,28,31,38)(26,32,27,35)(29,39,42,50)(30,40,43,51)(33,44,36,47)(34,45,37,48)(41,52,53,56)(46,54,49,55);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,21)(19,24)(25,29,31,42)(26,33,27,36)(28,39,38,50)(30,41,43,53)(32,44,35,47)(34,46,37,49)(40,52,51,56)(45,54,48,55);(25,30,31,43)(26,34,27,37)(28,40,38,51)(29,41,42,53)(32,45,35,48)(33,46,36,49)(39,52,50,56)(44,54,47,55)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,e^2=i,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, 7622876);