Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C22 | C2×D16 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2,3,4)(5,6)(7,18)(8,13)(9,27)(10,31)(11,17)(12,20)(14,29)(15,25)(16,26)(19,45)(21,51)(22,53)(23,46)(24,54)(28,42)(30,60)(32,39)(33,38)(34,47)(35,61)(36,50)(37,56)(40,66)(41,63)(43,58)(44,49)(48,65)(52,67)(55,64)(57,59)(62,68);(5,7)(6,12)(8,19)(9,28)(10,32)(11,23)(13,34)(14,39)(15,42)(16,44)(17,36)(18,27)(20,29)(21,25)(22,26)(24,55)(30,62)(31,35)(33,48)(37,51)(38,64)(40,67)(41,65)(43,68)(45,54)(46,60)(47,56)(49,66)(50,63)(52,58)(53,59)(57,61);(5,8,11,26)(6,13,17,16)(7,19,23,22)(9,24,30,57)(10,25,33,58)(12,34,36,44)(14,37,41,40)(15,38,43,31)(18,45,46,53)(20,47,50,49)(21,48,52,32)(27,54,60,59)(28,55,62,61)(29,56,63,66)(35,42,64,68)(39,51,65,67)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh76 conjugacy classes — view the full character table.
| maximal subgroup of | #151963, #151964, #152167, … |
| maximal normal subgroup of | #151963, #151964, #152167, … |
| maximal quotient of | #150056, #150119, #150134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6603);