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 | #62870 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,15)(3,16,23,55)(4,24,29,6)(5,12,33,48)(7,37,40,30)(8,14,35,26)(10,41,46,20)(11,47,36,28)(13,34,25,31)(17,56,58,60)(18,52,51,57)(19,53,59,43)(21,54,22,49)(27,32,39,50)(38,45,62,44)(42,64,61,63);(1,3)(2,10)(4,17)(5,18)(6,19)(7,38)(8,22)(9,23)(11,42)(12,43)(13,49)(14,45)(15,46)(16,37)(20,31)(21,35)(24,59)(25,54)(26,44)(27,63)(28,57)(29,58)(30,55)(32,56)(33,51)(34,41)(36,61)(39,64)(40,62)(47,52)(48,53)(50,60);(1,4)(2,11)(3,17)(5,30)(6,26)(7,39)(8,28)(9,29)(10,42)(12,31)(13,50)(14,24)(15,36)(16,51)(18,55)(19,44)(20,43)(21,52)(22,57)(23,58)(25,32)(27,40)(33,37)(34,48)(35,47)(38,64)(41,53)(45,59)(46,61)(49,60)(54,56)(62,63);(1,5)(2,12)(3,18)(4,25)(6,34)(7,11)(8,32)(9,33)(10,43)(13,29)(14,39)(15,48)(16,52)(17,54)(19,41)(20,59)(21,60)(22,56)(23,51)(24,31)(26,27)(28,30)(35,50)(36,40)(37,47)(38,42)(44,63)(45,64)(46,53)(49,58)(55,57)(61,62);(1,6,35,11)(2,4,26,47)(3,19,21,42)(5,31,50,40)(7,33,34,32)(8,36,9,24)(10,17,44,52)(12,13,27,30)(14,28,15,29)(16,53,54,64)(18,20,60,62)(22,61,23,59)(25,39,37,48)(38,51,41,56)(43,49,63,55)(45,57,46,58)Polycyclic presentation:
2^9|a^2=i,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, 6259621);