Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) = C4×#7622880 (order 2048) · Inn(G) = #120 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #120 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1850 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,11,26,37,8,13,27,38,5,22,25,39)(3,14,18,41,16,9,28,31,17,32,47,57,23,30,50,56)(4,10,20,34,19,35,48,59,21,36,49,60,24,33,51,58)(15,42,45,63,46,54,52,64,44,40,61,62,43,29,53,55);(1,3,6,18,7,16,26,28,8,17,27,47,5,23,25,50)(2,9,12,31,11,32,37,57,13,30,38,56,22,14,39,41)(4,15,20,45,19,46,48,52,21,44,49,61,24,43,51,53)(10,29,34,55,35,42,59,63,36,54,60,64,33,40,58,62);(1,4)(2,10)(3,15)(5,24)(6,20)(7,19)(8,21)(9,29)(11,35)(12,34)(13,36)(14,40)(16,46)(17,44)(18,45)(22,33)(23,43)(25,51)(26,48)(27,49)(28,52)(30,54)(31,55)(32,42)(37,59)(38,60)(39,58)(41,62)(47,61)(50,53)(56,64)(57,63)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^b=eh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93734, #93735, #135948, … |
| maximal normal subgroup of | #93734, #93735, #135948, … |
| maximal quotient of | #135812, #135825, #135837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5904);