Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | Aut(G) = #6344638 (order 512) · Inn(G) = #102 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #102 |
| commutator | C8 | C2×C16 |
| Frattini, cosocle | C4×C16 | C22 |
| socle | C2 | D4⋊C16 |
| prime radical | #135 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
59 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,12,7,11,28,39,8,13,29,42,4,21,23,51,9,14,30,43,24,40,54,63,6,31,27,58,25,44,55,64)(3,15,17,45,19,10,33,34,18,36,50,60,22,37,52,61,20,47,26,56,16,38,48,41,32,46,57,62,49,35,53,59);(1,3)(2,10)(4,22)(5,26)(6,32)(7,19)(8,18)(9,20)(11,36)(12,41)(13,37)(14,38)(15,44)(16,24)(17,30)(21,47)(23,53)(25,49)(27,50)(28,48)(29,57)(31,35)(33,54)(34,43)(39,62)(40,46)(42,59)(45,51)(52,55)(56,64)(58,61)(60,63)Polycyclic presentation:
2^8|a^2=d,c^2=e,d^2=f,e^2=h,f^2=g,g^2=h,b^a=bc,c^a=ce,e^a=eh,c^b=ceh,d^b=dh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7382, #150053, #150107, … |
| maximal normal subgroup of | C2×#7382, #150053, #150107, … |
| maximal quotient of | #93746, #93754, #93755, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 370);