Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13392 |
| automorphism | Aut(G) = #72312248 (order 2048) · Inn(G) = #482 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #482 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2033 |
| 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 | |
193 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 193 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)(3,13)(4,19)(5,25)(6,28)(7,12)(8,15)(9,22)(10,27)(11,26)(14,44)(16,33)(17,34)(18,35)(20,54)(21,52)(23,53)(24,40)(29,39)(30,37)(31,58)(32,60)(36,57)(38,56)(41,59)(42,46)(43,49)(45,61)(47,63)(48,62)(50,51)(55,64);(1,3)(2,8)(4,20)(5,17)(6,29)(7,18)(9,32)(10,35)(11,39)(12,34)(13,25)(14,23)(15,27)(16,31)(19,51)(21,45)(22,46)(24,50)(26,47)(28,43)(30,48)(33,41)(36,64)(37,42)(38,62)(40,44)(49,59)(52,54)(53,61)(55,56)(57,60)(58,63);(1,4,5,21,7,23,27,24)(2,9,10,36,12,38,25,37)(3,14,15,45,18,20,17,50)(6,30,26,56,31,57,59,22)(8,32,13,42,34,62,35,64)(11,40,28,53,41,52,58,19)(16,48,47,46,29,60,49,55)(33,54,43,61,39,44,63,51)Polycyclic presentation:
2^8|c^2=d,d^2=h,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=dh,e^b=eh,e^c=egh,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13392, #499029, #499595, … |
| maximal normal subgroup of | C2×#13392, #499029, #499595, … |
| maximal quotient of | #144686, #144687, #144778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6380);