Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352341 |
| 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 | #62872 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,16)(3,17,22,53)(4,11,26,33)(5,12,30,25)(6,13,34,44)(7,35,37,24)(8,15,32,46)(10,38,42,21)(14,31,27,28)(18,19,51,50)(20,52,56,40)(23,48,58,55)(29,45,49,47)(36,43,62,63)(39,61,60,54)(41,57,64,59);(1,3)(2,10)(4,18)(5,19)(6,20)(7,36)(8,23)(9,22)(11,39)(12,40)(13,41)(14,48)(15,43)(16,42)(17,35)(21,28)(24,53)(25,52)(26,51)(27,55)(29,57)(30,50)(31,38)(32,58)(33,60)(34,56)(37,62)(44,64)(45,61)(46,63)(47,54)(49,59);(1,4,8,13)(2,11,15,34)(3,18,23,41)(5,27,29,24)(6,16,33,46)(7,12,28,45)(9,26,32,44)(10,39,43,56)(14,49,35,30)(17,50,48,59)(19,55,57,53)(20,42,60,63)(21,61,36,40)(22,51,58,64)(25,31,47,37)(38,54,62,52);(1,5)(2,12)(3,19)(4,24)(6,31)(7,11)(8,29)(9,30)(10,40)(13,27)(14,44)(15,45)(16,25)(17,51)(18,53)(20,38)(21,56)(22,50)(23,57)(26,35)(28,34)(32,49)(33,37)(36,39)(41,55)(42,52)(43,61)(46,47)(48,64)(54,63)(58,59)(60,62);(1,6,32,11)(2,13,46,26)(3,20,58,39)(4,16,44,15)(5,28,49,37)(7,30,31,29)(8,33,9,34)(10,41,63,51)(12,14,47,24)(17,52,55,61)(18,42,64,43)(19,21,59,62)(22,56,23,60)(25,27,45,35)(36,50,38,57)(40,48,54,53)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259622);