Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #333127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3253 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,7,12,6,8)(3,9,13,23,15,24,20,21)(4,16,17,25,18,26,19,10)(14,28,29,32,30,31,27,22)(33,34,39,43,38,44,37,42)(35,45,48,40,49,60,47,55)(36,41,53,58,52,59,51,57)(46,56,62,54,63,61,50,64);(2,8)(3,13)(4,17)(5,6)(10,25)(11,12)(15,20)(18,19)(21,23)(22,28)(27,29)(31,32)(33,35)(34,40)(36,50)(37,48)(38,49)(39,47)(41,56)(42,60)(43,45)(44,55)(46,51)(52,62)(53,63)(54,57)(58,64)(59,61);(1,3)(2,9)(4,14)(5,13)(6,20)(7,15)(8,21)(10,22)(11,23)(12,24)(16,28)(17,29)(18,30)(19,27)(25,32)(26,31)(33,35)(34,40)(36,46)(37,47)(38,49)(39,48)(41,54)(42,45)(43,60)(44,55)(50,51)(52,63)(53,62)(56,57)(58,61)(59,64);(1,4,7,18)(2,10,12,25)(3,14,15,30)(5,19,6,17)(8,16,11,26)(9,22,24,32)(13,27,20,29)(21,28,23,31)(33,36,38,52)(34,41,44,59)(35,46,49,63)(37,53,39,51)(40,54,55,64)(42,58,43,57)(45,61,60,56)(47,62,48,50)Polycyclic presentation:
2^9|a^2=ei,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 240345);