Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62906 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2681 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,30)(3,16,31,33)(4,14,29,37)(5,13,23,52)(6,27,22,11)(7,15,8,12)(10,32,55,17)(18,43,47,57)(19,42,20,44)(21,41,50,60)(24,53,51,38)(25,40,26,39)(28,34,46,36)(35,56,64,45)(48,61,49,62)(54,58,63,59);(1,3)(2,10)(4,17)(5,18)(6,28)(7,20)(8,19)(9,31)(11,33)(12,34)(13,35)(14,42)(15,36)(16,27)(21,45)(22,46)(23,47)(24,54)(25,49)(26,48)(29,32)(30,55)(37,44)(38,57)(39,58)(40,59)(41,61)(43,53)(50,56)(51,63)(52,64)(60,62);(1,4,9,29)(2,11,30,27)(3,17,31,32)(5,21,23,50)(6,7,22,8)(10,33,55,16)(12,37,15,14)(13,38,52,53)(18,45,47,56)(19,28,20,46)(24,25,51,26)(34,44,36,42)(35,57,64,43)(39,60,40,41)(48,54,49,63)(58,62,59,61);(2,12)(5,23)(10,34)(11,37)(13,40)(14,27)(15,30)(16,42)(18,47)(21,50)(24,51)(25,26)(33,44)(35,59)(36,55)(38,41)(39,52)(43,62)(45,56)(48,49)(53,60)(54,63)(57,61)(58,64);(1,5,7,25)(2,13,15,40)(3,18,20,49)(4,21,22,51)(6,24,29,50)(8,26,9,23)(10,35,36,59)(11,38,14,41)(12,39,30,52)(16,43,44,62)(17,45,46,63)(19,48,31,47)(27,53,37,60)(28,54,32,56)(33,57,42,61)(34,58,55,64)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260491);