Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343678 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(2,8,11,29)(4,18)(5,23)(9,27,33,54)(10,35,37,58)(12,30,13,24)(15,40)(16,44)(19,46)(21,50)(22,48)(25,26)(28,38,57,39)(31,45,34,55)(32,60,61,56)(36,62,59,53)(41,52)(43,64)(47,51)(49,63);(1,2,6,12)(3,9,17,34)(4,10,22,38)(5,24,23,29)(7,13,20,11)(8,26,30,25)(14,31,42,33)(15,32,19,36)(16,45,44,54)(18,39,48,37)(21,35,51,57)(27,52,55,41)(28,50,58,47)(40,59,46,61)(43,60,49,62)(53,64,56,63);(1,3,6,17)(2,9,12,34)(4,19,22,15)(5,16,23,44)(7,14,20,42)(8,27,30,55)(10,36,38,32)(11,33,13,31)(18,46,48,40)(21,49,51,43)(24,45,29,54)(25,41,26,52)(28,56,58,53)(35,62,57,60)(37,59,39,61)(47,64,50,63);(3,14)(9,31)(15,40)(16,41)(17,42)(19,46)(27,45)(32,59)(33,34)(36,61)(43,63)(44,52)(49,64)(53,60)(54,55)(56,62);(1,4,20,48)(2,10,11,37)(3,15,42,46)(5,21,26,50)(6,22,7,18)(8,28,29,57)(9,32,33,61)(12,38,13,39)(14,40,17,19)(16,43,52,64)(23,51,25,47)(24,35,30,58)(27,53,54,62)(31,59,34,36)(41,63,44,49)(45,60,55,56)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=hi,f^2=i,b^a=bf,e^a=ei,f^a=fh,f^b=fi,g^b=ghi,d^c=di,e^c=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250895);