Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | Aut(G) = C2×#6372601 (order 1024) · Inn(G) = C23 · Out(G) = C22×#131 |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C23 |
| commutator, socle | C22 | C22×C16 |
| prime radical | C23 | C2×C16 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,17,13,18,25,35,14,19,26,36,28,38,47,57,15,20,27,37,29,39,48,58,30,40,49,59,50,60,65,70)(11,16,22,32,23,33,44,54,24,34,45,55,46,56,64,69,21,31,41,51,42,52,61,66,43,53,62,67,63,68,71,72);(2,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,24)(15,21)(17,32)(18,33)(19,34)(20,31)(25,44)(26,45)(27,41)(28,46)(29,42)(30,43)(35,54)(36,55)(37,51)(38,56)(39,52)(40,53)(47,64)(48,61)(49,62)(50,63)(57,69)(58,66)(59,67)(60,68)(65,71)(70,72);(1,3,4,7)(2,6,5,8)(11,21)(16,31)(22,41)(23,42)(24,43)(32,51)(33,52)(34,53)(44,61)(45,62)(46,63)(54,66)(55,67)(56,68)(64,71)(69,72)Polycyclic presentation:
2^8|a^2=e,c^2=d,e^2=f,f^2=g,g^2=h,b^a=bd,c^b=ch112 conjugacy classes — view the full character table.
| maximal subgroup of | #94003, #94004, #94210, … |
| maximal normal subgroup of | #94003, #94004, #94210, … |
| maximal quotient of | #151878, #151886, #151976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6629);