Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7266801 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #46996 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #23896 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
1545 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,13)(4,9)(5,10)(6,11)(7,12)(8,17)(14,26)(15,33)(16,28)(18,30)(19,31)(20,32)(21,27)(22,35)(23,36)(24,38)(25,39)(29,42)(34,48)(37,45)(40,52)(41,56)(43,55)(44,53)(46,54)(47,57)(49,58)(50,59)(51,60)(61,64)(62,63);(1,3)(2,8,12,13)(4,14)(5,21,23,15)(6,16)(7,17)(9,29,32,26)(10,33)(11,37,39,28)(18,46,48,40)(19,41)(20,42)(22,49,51,43)(24,44)(25,45)(27,36)(30,52)(31,53,38,56)(34,54)(35,55)(47,62,50,61)(57,64)(58,60)(59,63);(1,4,7,20)(2,9,12,32)(3,14,17,42)(5,18,23,48)(6,24,25,19)(8,26,13,29)(10,34,36,30)(11,38,39,31)(15,40,21,46)(16,44,45,41)(22,50,51,47)(27,54,33,52)(28,56,37,53)(35,57,60,59)(43,62,49,61)(55,63,58,64);(1,5)(2,10)(3,15)(4,18)(6,22)(7,23)(8,27)(9,30)(11,35)(12,36)(13,33)(14,40)(16,43)(17,21)(19,47)(20,48)(24,50)(25,51)(26,52)(28,55)(29,54)(31,57)(32,34)(37,58)(38,59)(39,60)(41,61)(42,46)(44,62)(45,49)(53,63)(56,64);(1,6,7,25)(2,11,12,39)(3,16,17,45)(4,19,20,24)(5,22,23,51)(8,28,13,37)(9,31,32,38)(10,35,36,60)(14,41,42,44)(15,43,21,49)(18,47,48,50)(26,53,29,56)(27,55,33,58)(30,57,34,59)(40,61,46,62)(52,63,54,64)Polycyclic presentation:
2^9|b^2=fgh,c^2=i,e^2=i,b^a=bf,c^a=cg,e^a=ef,h^a=hi,c^b=ch,d^b=df,e^b=eh,f^b=fi,g^b=gi,d^c=dh,e^c=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7173999);