Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1303 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)(3,7)(4,6)(5,8)(25,26,30,34)(27,31,36,46)(28,38,40,50)(29,42,44,33)(32,43,49,41)(35,51,39,56)(37,54,53,47)(45,55,48,52);(1,3,5,4)(2,6,8,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27)(26,31)(28,39)(29,37)(30,36)(32,48)(33,47)(34,46)(35,40)(38,56)(41,55)(42,54)(43,52)(44,53)(45,49)(50,51);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28)(26,32)(27,35)(29,43)(30,40)(31,45)(33,50)(34,49)(36,39)(37,55)(38,42)(41,44)(46,48)(47,56)(51,54)(52,53);(1,3,5,4)(2,7,8,6)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(27,36)(28,40)(31,46)(32,49)(37,53)(38,50)(41,43)(47,54);(1,4,5,3)(2,6,8,7)(10,16)(14,17)(15,24)(22,23)(27,36)(31,46)(35,39)(37,53)(45,48)(47,54)(51,56)(52,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=fhi,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=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6437604);