Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#2258 (order 256) · Inn(G) = C22 · Out(G) = C22×C16 |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C1 |
| modular maximal cyclic | M256 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C64 | C22 |
| commutator, socle | C2 | C2×C64 |
| prime radical | C22 | C64 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C64 ◁ G | |
| LECS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128);(2,66)(4,68)(6,70)(8,72)(10,74)(12,76)(14,78)(16,80)(18,82)(20,84)(22,86)(24,88)(26,90)(28,92)(30,94)(32,96)(34,98)(36,100)(38,102)(40,104)(42,106)(44,108)(46,110)(48,112)(50,114)(52,116)(54,118)(56,120)(58,122)(60,124)(62,126)(64,128)Polycyclic presentation:
2^8|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,g^2=h,b^a=bh160 conjugacy classes — view the full character table.
| maximal subgroup of | #94815, #94817, #94818, … |
| maximal normal subgroup of | #94815, #94817, #94818, … |
| maximal quotient of | #94814, C22⋊C128, C4⋊C128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 538);