Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #236803 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,16,24)(4,9,20,27)(5,11,6,10)(8,17,23,15)(14,29,30,25)(18,32,31,22)(19,28,21,26)(33,34)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,59)(45,54)(46,58)(47,55)(50,56)(52,57)(53,60)(61,63)(62,64);(1,3)(2,8)(4,18)(5,17)(6,15)(7,16)(9,25)(10,24)(11,13)(12,23)(14,19)(20,31)(21,30)(22,26)(27,29)(28,32)(33,35,39,47)(34,40,43,55)(36,44,50,61)(37,46,38,45)(41,53,57,64)(42,58,51,54)(48,63,56,59)(49,62,52,60);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,28)(12,27)(13,25)(15,18)(16,30)(17,31)(23,32)(24,29)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,56)(43,57)(45,60)(46,62)(47,61)(48,51)(54,63)(55,64)(58,59);(33,37,39,38)(34,42,43,51)(35,45,47,46)(36,49,50,52)(40,54,55,58)(41,56,57,48)(44,60,61,62)(53,63,64,59)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,g^b=gh,e^c=ei,f^c=fi,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 144016);