Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13661 |
| automorphism | Aut(G) = C24×#12067 (order 4096) · Inn(G) = C2×D16 · Out(G) = C23×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
144 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 68 points:
(1,2)(3,4)(5,6,11,17)(7,18,23,49)(8,13,26,16)(9,27,30,61)(10,31,33,39)(12,34,37,20)(14,40,43,29)(15,44,46,25)(19,48,22,55)(21,53,54,67)(24,56,59,60)(28,45,63,68)(32,41,51,66)(35,52,47,50)(36,57,64,62)(38,65,42,58);(5,7)(6,12)(8,19)(9,28)(10,32)(11,23)(13,35)(14,41)(15,45)(16,47)(17,37)(18,27)(20,29)(21,25)(22,26)(24,57)(30,63)(31,36)(33,51)(34,40)(38,53)(39,64)(42,67)(43,66)(44,54)(46,68)(48,56)(49,61)(50,58)(52,65)(55,60)(59,62);(2,4)(5,8,11,26)(6,13,17,16)(7,19,23,22)(9,24,30,59)(10,25,33,44)(12,35,37,47)(14,38,43,42)(15,39,46,31)(18,48,49,55)(20,50,34,52)(21,51,54,32)(27,56,61,60)(28,57,63,62)(29,58,40,65)(36,45,64,68)(41,53,66,67)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=fg,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13666, #512826, #512829, … |
| maximal normal subgroup of | C2×#13666, #512826, #512829, … |
| maximal quotient of | #151175, #151287, #151485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6654);