Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,8,4,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,30,36)(27,37,40,32)(28,34,43,52)(29,35,31,46)(33,45,53,42)(38,54,41,48)(39,50,44,49)(47,56,51,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,30,40)(26,32,36,37)(28,41,43,38)(29,44,31,39)(33,51,53,47)(34,54,52,48)(35,50,46,49)(42,55,45,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,43)(26,33,36,53)(27,38,40,41)(29,42,31,45)(32,47,37,51)(34,35,52,46)(39,55,44,56)(48,50,54,49);(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,42)(32,48)(33,46)(35,53)(36,52)(37,54)(39,56)(40,41)(44,55)(47,49)(50,51);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(25,29,30,31)(26,35,36,46)(27,39,40,44)(28,42,43,45)(32,49,37,50)(33,52,53,34)(38,55,41,56)(47,48,51,54)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6417892);