Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = C22×#72224639 (order 4096) · Inn(G) = D32 · Out(G) = C22×C16 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D32 |
| commutator | C32 | C2×C4 |
| Frattini, cosocle | C2×C32 | C22 |
| prime radical | C2×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ G | |
| LECS | G ▷ C2×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
49 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)(7,12)(8,23)(9,27)(10,30)(11,17)(13,18)(14,19)(15,24)(16,28)(20,40)(21,42)(22,34)(25,58)(26,56)(29,60)(31,44)(32,43)(33,51)(35,50)(36,53)(37,46)(38,59)(39,49)(41,57)(45,61)(47,55)(48,54)(52,65)(62,68)(63,67)(64,66);(6,12)(7,18)(8,19)(9,24)(10,28)(13,14)(15,40)(16,42)(17,34)(20,50)(21,53)(22,46)(23,31)(25,59)(26,49)(27,32)(29,57)(30,33)(35,55)(36,54)(37,39)(38,61)(41,65)(43,44)(45,51)(47,48)(52,68)(56,63)(58,62)(60,64)(66,67)Polycyclic presentation:
2^8|a^2=d,c^2=ef,e^2=fg,f^2=gh,g^2=h,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,c^b=ce,e^b=ef,f^b=fg,g^b=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7537, #153555, #153556, … |
| maximal normal subgroup of | C2×#7537, #153555, #153556, … |
| maximal quotient of | #94607, #94690, #94734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 525);