Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #447 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1852 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 73 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:
(1,2,6,13,8,14,7,12)(3,15,18,42,19,43,30,37)(4,20,23,36,24,40,22,10)(5,25,27,39,29,38,28,11)(9,26,33,49,34,57,32,17)(16,44,47,62,48,60,46,41)(21,52,55,61,56,35,54,51)(31,45,50,53,59,64,58,63);(1,3,6,18,8,19,7,30)(2,9,13,33,14,34,12,32)(4,16,23,47,24,48,22,46)(5,26,27,49,29,57,28,17)(10,31,20,50,36,59,40,58)(11,37,25,15,39,42,38,43)(21,53,55,64,56,63,54,45)(35,60,51,41,52,44,61,62);(1,4)(2,10)(3,16)(5,21)(6,23)(7,22)(8,24)(9,31)(11,35)(12,40)(13,20)(14,36)(15,41)(17,45)(18,47)(19,48)(25,51)(26,53)(27,55)(28,54)(29,56)(30,46)(32,58)(33,50)(34,59)(37,60)(38,61)(39,52)(42,44)(43,62)(49,64)(57,63)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12927, #494259, #495719, … |
| maximal normal subgroup of | C2×#12927, #494259, #495719, … |
| maximal quotient of | #135820, #135821, #135843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5915);