Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #2263 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D16 |
| commutator | C16 | C22×C4 |
| Frattini, cosocle | C2×C16 | C23 |
| socle | C2 | C2×#357 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
145 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 145 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,5,10,7,13,26,12)(3,14,17,44,15,43,19,50)(4,20,23,51,25,53,58,52)(6,27,22,11,29,39,57,41)(8,30,32,45,31,47,42,16)(9,34,36,59,38,61,37,24)(18,49,48,62,46,63,28,35)(21,40,55,60,54,64,56,33);(1,3)(2,8)(4,21)(5,17)(6,28)(7,15)(9,35)(10,32)(11,40)(12,42)(13,31)(14,20)(16,24)(18,22)(19,26)(23,55)(25,54)(27,33)(29,48)(30,34)(36,49)(37,63)(38,62)(39,60)(41,64)(43,53)(44,51)(45,59)(46,57)(47,61)(50,52)(56,58);(3,15)(8,31)(14,43)(16,45)(17,19)(18,46)(21,54)(28,48)(30,47)(32,42)(33,60)(35,62)(40,64)(44,50)(49,63)(55,56)Polycyclic presentation:
2^8|a^2=e,d^2=fg,e^2=h,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 | #151971, #151973, #152166, … |
| maximal normal subgroup of | #151971, #151973, #152166, … |
| maximal quotient of | #150057, #150058, #150059, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6608);