Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13312 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #469 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2034 |
| prime radical | #3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 53 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)(3,14)(4,20)(5,12)(6,28)(7,13)(8,16)(9,23)(10,27)(11,26)(15,46)(17,33)(18,35)(19,34)(21,54)(22,55)(24,41)(25,53)(29,40)(30,39)(31,42)(32,51)(36,62)(37,58)(38,59)(43,57)(44,48)(45,52)(47,64)(49,63)(50,61)(56,60);(1,3)(2,8)(4,21)(5,18)(6,29)(7,19)(9,36)(10,34)(11,40)(12,14)(13,35)(15,24)(16,27)(17,31)(20,46)(22,49)(23,56)(25,47)(26,50)(28,45)(30,48)(32,37)(33,43)(38,44)(39,60)(41,64)(42,61)(51,59)(52,57)(53,63)(54,55)(58,62);(1,4,7,25)(2,9,13,38)(3,15)(5,24,27,22)(6,30,31,58)(8,32)(10,39,12,37)(11,41,43,55)(14,44)(16,21)(17,51)(18,47)(19,49)(20,42,53,28)(23,57,59,26)(29,56)(33,46)(34,36)(35,60)(40,63)(45,64)(48,52)(50,62)(54,61)Polycyclic presentation:
2^8|c^2=dg,d^2=h,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,g^a=gh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13406, #499045, #499508, … |
| maximal normal subgroup of | C2×#13406, #499045, #499508, … |
| maximal quotient of | #144720, #144721, #144748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6394);