Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | Aut(G) = #72314138 (order 4096) · Inn(G) = #469 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2076 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
138 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,25,11,41,38,7,13,27,40,6,12,26,39)(3,8,15,30,16,42,48,63,18,32,50,60,17,31,49,59)(4,19,22,35,28,53,57,34,21,52,56,36,23,37,24,9)(14,43,45,61,20,54,55,58,44,62,64,33,46,51,47,29);(1,3)(2,8)(4,20)(5,15)(6,17)(7,18)(9,33)(10,30)(11,42)(12,31)(13,32)(14,23)(16,25)(19,51)(21,46)(22,55)(24,45)(26,49)(27,50)(28,44)(29,35)(34,61)(36,58)(37,62)(38,63)(39,59)(40,60)(41,48)(43,53)(47,56)(52,54)(57,64);(1,4,6,28,7,21,25,23)(2,9,12,35,13,34,11,36)(3,14,17,20,18,44,16,46)(5,24,26,22,27,57,41,56)(8,29,31,61,32,58,42,33)(10,37,39,19,40,53,38,52)(15,47,49,45,50,55,48,64)(30,51,59,43,60,54,63,62)Polycyclic presentation:
2^8|a^2=e,c^2=dg,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=eg,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93788, #93789, #151493, … |
| maximal normal subgroup of | #93788, #93789, #151493, … |
| maximal quotient of | #146023, #146033, #146057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6487);