Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | Aut(G) = #72287333 (order 8192) · Inn(G) = D32 · Out(G) = D4×C16 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | D32 |
| commutator | C32 | C2×C4 |
| Frattini, cosocle | C2×C32 | C22 |
| 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 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2,3,4)(6,12)(7,18)(8,19)(9,23)(10,26)(13,22)(14,37)(15,41)(16,44)(17,32)(20,51)(21,36)(24,55)(25,50)(27,54)(28,29)(30,38)(31,42)(33,58)(34,35)(39,67)(40,62)(43,61)(45,46)(47,52)(48,49)(53,64)(56,65)(57,63)(59,68)(60,66);(5,6,13,33,8,14,34,61,23,38,64,67,49,57,52,44,10,16,36,63,24,39,20,30,27,43,46,37,50,58,7,12,11,17,18,29,25,40,45,41,54,66,51,65,55,68,21,31,26,42,47,59,48,56,53,60,9,15,35,62,19,28,22,32)Polycyclic presentation:
2^8|a^2=d,b^2=ceh,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=gh70 conjugacy classes — view the full character table.
| maximal subgroup of | #94835, #94843, C2×#7546, … |
| maximal normal subgroup of | #94835, #94843, C2×#7546, … |
| maximal quotient of | #94750, #94761, #94843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 534);