Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| prime radical | C23 | D16 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
145 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)(19,46)(21,52)(22,48)(23,47)(24,57)(28,43)(30,45)(32,40)(33,39)(34,49)(35,62)(36,55)(37,51)(38,59)(41,68)(42,50)(44,53)(54,66)(56,67)(58,65)(60,61)(63,64);(5,7,11,22)(6,12,17,37)(8,23,26,19)(9,28,30,63)(10,32,33,56)(13,36,16,34)(14,40,42,67)(15,43,44,64)(18,45,48,27)(20,50,51,29)(21,53,54,25)(24,58,60,62)(31,35,39,65)(38,52,41,66)(46,57,47,61)(49,59,55,68);(5,8,11,26)(6,13,17,16)(7,19,22,23)(9,24,30,60)(10,25,33,53)(12,34,37,36)(14,38,42,41)(15,39,44,31)(18,46,48,47)(20,49,51,55)(21,32,54,56)(27,57,45,61)(28,62,63,58)(29,59,50,68)(35,64,65,43)(40,66,67,52)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,c^b=ch,d^b=df,f^b=fg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #151970, #152170, #152177, … |
| maximal normal subgroup of | #151970, #152170, #152177, … |
| maximal quotient of | #150059, #150122, #150137, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6606);