Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D16 |
| commutator | C16 | C42 |
| Frattini, cosocle | C22×C16 | C22 |
| socle | C22 | #357 |
| prime radical | C23 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
85 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(5,6)(7,19)(8,25)(9,31)(10,34)(11,35)(12,18)(13,20)(14,28)(15,41)(16,26)(17,27)(21,54)(22,36)(23,57)(24,48)(29,44)(30,43)(32,61)(33,58)(37,56)(38,65)(39,53)(40,60)(42,59)(45,51)(46,49)(47,50)(52,62)(55,64)(63,67)(66,68);(5,7,9,21,32,55,8,20,27,50,60,68,26,49,41,65,12,24,33,56,10,22,30,53,11,23,28,51,59,67,29,52)(6,13,15,36,42,54,40,48,35,62,25,46,34,45,31,47,18,39,44,64,16,37,14,19,17,38,43,63,61,66,58,57)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,e^2=fh,f^2=gh,g^2=h,b^a=bc,c^a=cf,e^a=efg,f^a=fg,g^a=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7473, #151772, C4×#1417, … |
| maximal normal subgroup of | C2×#7473, #151772, C4×#1417, … |
| maximal quotient of | #94022, #94044, #94045, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 461);