Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1280 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,36)(27,38,41,32)(28,43,45,34)(29,37,31,35)(33,46,51,44)(39,48,55,53)(40,49,42,50)(47,54,52,56);(3,4)(6,7)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27,30,41)(26,32,36,38)(28,39,45,55)(29,40,31,42)(33,47,51,52)(34,48,43,53)(35,49,37,50)(44,54,46,56);(1,3,5,4)(2,6,8,7)(25,28)(26,33)(27,39)(29,44)(30,45)(31,46)(32,47)(34,37)(35,43)(36,51)(38,52)(40,54)(41,55)(42,56)(48,50)(49,53);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,34)(27,39)(29,46)(30,45)(31,44)(32,48)(33,37)(35,51)(36,43)(38,53)(40,56)(41,55)(42,54)(47,50)(49,52);(1,4,5,3)(2,7,8,6)(10,16)(14,17)(15,24)(22,23)(25,29,30,31)(26,35,36,37)(27,40,41,42)(28,44,45,46)(32,49,38,50)(33,43,51,34)(39,54,55,56)(47,53,52,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7141272);