Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13691 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2206 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C2 | C2×#2206 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14)(4,20)(5,12)(6,26)(7,28)(8,32)(9,15)(10,17)(11,23)(13,30)(16,45)(18,49)(19,44)(21,53)(22,54)(24,55)(25,58)(27,35)(29,43)(31,61)(33,41)(34,60)(36,56)(37,59)(38,50)(39,57)(40,51)(42,48)(46,52)(47,64)(62,63);(1,3,6,17,9,19,5,8)(2,10,12,14,15,32,26,44)(4,16,22,46,24,47,21,33)(7,18,27,48,31,51,25,30)(11,34,36,43,39,37,38,63)(13,35,40,28,42,58,49,61)(20,52,53,45,55,41,54,64)(23,29,50,60,57,62,56,59);(1,4)(2,11)(3,16)(5,21)(6,22)(7,29)(8,33)(9,24)(10,34)(12,36)(13,41)(14,43)(15,39)(17,46)(18,50)(19,47)(20,28)(23,30)(25,59)(26,38)(27,60)(31,62)(32,37)(35,54)(40,64)(42,52)(44,63)(45,49)(48,57)(51,56)(53,58)(55,61);(1,5,9,6)(2,12,15,26)(3,8,19,17)(4,21,24,22)(7,25,31,27)(10,14,32,44)(11,36,39,38)(13,40,42,49)(16,33,47,46)(18,30,51,48)(20,53,55,54)(23,56,57,50)(28,58,61,35)(29,59,62,60)(34,43,37,63)(41,64,52,45)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fg,g^a=gh,h^a=hi,f^c=fg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72225337, #72225339, #72225340, … |
| maximal normal subgroup of | #72225337, #72225339, #72225340, … |
| maximal quotient of | #72225449 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420124);