Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1264 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,33,30,36,29,35)(27,32,42,47,37,46,39,44)(28,34,43,48,40,49,41,50)(38,45,53,56,51,55,52,54);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,37)(26,32,36,46)(28,38,40,51)(29,39,31,42)(33,47,35,44)(34,45,49,55)(41,52,43,53)(48,56,50,54);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(27,37)(29,31)(32,44)(34,48)(35,36)(38,51)(41,43)(45,54)(46,47)(49,50)(55,56);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28,30,40)(26,34,36,49)(27,38,37,51)(29,41,31,43)(32,45,46,55)(33,48,35,50)(39,52,42,53)(44,54,47,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(27,37)(28,40)(32,46)(34,49)(39,42)(41,43)(44,47)(48,50)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=f,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6775614);