Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13074 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1893 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 229 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)(3,13)(4,19,21,9)(5,23)(6,28)(7,11)(8,16)(10,26)(12,25)(14,36)(15,37,44,57)(17,40)(18,50)(20,53,24,55)(22,34,54,52)(27,29)(30,43,51,46)(31,41)(32,48)(33,42,35,56)(38,49)(39,47)(45,63,62,59)(58,64,60,61);(1,3,14,41,6,18,5,16,7,17,26,49,25,47,27,48)(2,8,29,50,12,31,10,13,11,32,23,39,28,38,36,40)(4,15,42,61,22,45,24,46,21,44,56,64,54,62,20,43)(9,30,55,63,34,58,35,37,19,51,53,59,52,60,33,57);(1,4)(2,9)(3,15)(5,24)(6,22)(7,21)(8,30)(10,35)(11,19)(12,34)(13,37)(14,42)(16,46)(17,44)(18,45)(20,27)(23,53)(25,54)(26,56)(28,52)(29,55)(31,58)(32,51)(33,36)(38,60)(39,59)(40,57)(41,61)(43,48)(47,62)(49,64)(50,63)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=eh,d^c=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130238, #130239, #130285, … |
| maximal normal subgroup of | #130238, #130239, #130285, … |
| maximal quotient of | #137620, #137660, #137985, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6063);