Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | Aut(G) = #72313012 (order 4096) · Inn(G) = #418 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #418 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1786 |
| prime radical | #3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
443 conjugacy classes of subgroups; only the 85 normal subgroups are shown.
This group has 443 conjugacy classes of subgroups — too many to draw. Only its 85 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:
(1,2,6,13,8,12,7,30)(3,14,18,43,17,44,31,39)(4,19,21,38,22,10,29,37)(5,23,26,41,28,55,27,11)(9,24,33,50,35,57,34,16)(15,42,47,64,46,60,48,63)(20,52,54,40,25,58,59,36)(32,53,61,45,51,56,62,49);(1,3)(2,9)(4,15)(5,24)(6,18)(7,31)(8,17)(10,32)(11,39)(12,35)(13,33)(14,23)(16,27)(19,51)(20,53)(21,47)(22,46)(25,56)(26,50)(28,57)(29,48)(30,34)(36,63)(37,61)(38,62)(40,64)(41,43)(42,52)(44,55)(45,54)(49,59)(58,60);(1,4)(2,10)(3,15)(5,25)(6,29)(7,21)(8,22)(9,32)(11,40)(12,19)(13,38)(14,42)(16,49)(17,46)(18,48)(20,28)(23,52)(24,56)(26,54)(27,59)(30,37)(31,47)(33,62)(34,61)(35,51)(36,41)(39,64)(43,63)(44,60)(45,50)(53,57)(55,58)Polycyclic presentation:
2^8|a^2=f,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dg,e^b=eh,g^b=gh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139571, #139572, #139575, … |
| maximal normal subgroup of | #139571, #139572, #139575, … |
| maximal quotient of | #129507, #129558, #129776, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5734);