Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431743 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#127 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2397 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2679 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,11,7,12,24,35)(3,13,16,27,17,34,23,8)(4,9,20,31,19,30,46,55)(5,10,22,33,21,32,48,56)(14,38,41,25,40,36,47,28)(15,37,43,51,42,57,49,26)(18,29,44,53,45,54,63,64)(39,59,60,50,61,58,62,52);(1,3)(2,8)(4,14)(5,15)(6,23)(7,17)(9,28)(10,26)(11,34)(12,27)(13,35)(16,24)(18,39)(19,40)(20,47)(21,42)(22,49)(25,30)(29,52)(31,36)(32,51)(33,57)(37,56)(38,55)(41,46)(43,48)(44,62)(45,61)(50,54)(53,58)(59,64)(60,63);(1,4)(2,9)(3,14)(5,18)(6,20)(7,19)(8,25)(10,29)(11,31)(12,30)(13,36)(15,39)(16,41)(17,40)(21,45)(22,44)(23,47)(24,46)(26,50)(27,28)(32,54)(33,53)(34,38)(35,55)(37,58)(42,61)(43,60)(48,63)(49,62)(51,52)(56,64)(57,59);(1,5,6,22,7,21,24,48)(2,10,11,33,12,32,35,56)(3,15,23,49,17,42,16,43)(4,18,20,44,19,45,46,63)(8,26,34,57,27,51,13,37)(9,29,31,53,30,54,55,64)(14,39,47,62,40,61,41,60)(25,50,38,59,28,52,36,58);(4,19)(5,21)(9,30)(10,32)(14,40)(15,42)(20,46)(22,48)(25,28)(26,51)(31,55)(33,56)(36,38)(37,57)(41,47)(43,49)Polycyclic presentation:
2^9|a^2=f,d^2=fg,f^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,f^b=fi,g^b=gi,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72285536, #72285582, #72286055, … |
| maximal normal subgroup of | #72285536, #72285582, #72286055, … |
| maximal quotient of | C2×#6432583 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339779);