Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #22455 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#578 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 643 conjugacy classes of subgroups — too many to draw. Only its 197 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 — 4 generators on 64 points:
(1,2,7,12,8,13,6,26)(3,14,17,9,18,33,27,32)(4,19,21,35,22,10,29,36)(5,11,24,40,25,39,28,38)(15,43,46,30,47,41,45,50)(16,42,48,31,49,37,23,56)(20,51,54,60,55,34,53,59)(44,58,52,57,62,61,63,64);(1,3)(2,9)(4,15)(5,23)(6,27)(7,17)(8,18)(10,30)(11,37)(12,33)(13,32)(14,26)(16,24)(19,50)(20,52)(21,46)(22,47)(25,48)(28,49)(29,45)(31,38)(34,58)(35,43)(36,41)(39,42)(40,56)(44,53)(51,61)(54,62)(55,63)(57,59)(60,64);(1,4,8,22)(2,10,13,19)(3,15,18,47)(5,20,25,55)(6,21,7,29)(9,30,32,50)(11,34,39,51)(12,35,26,36)(14,41,33,43)(16,44,49,62)(17,45,27,46)(23,52,48,63)(24,53,28,54)(31,57,56,64)(37,58,42,61)(38,59,40,60);(1,5)(2,11)(3,16)(4,20)(6,28)(7,24)(8,25)(9,31)(10,34)(12,40)(13,39)(14,42)(15,44)(17,48)(18,49)(19,51)(21,54)(22,55)(23,27)(26,38)(29,53)(30,57)(32,56)(33,37)(35,60)(36,59)(41,61)(43,58)(45,63)(46,52)(47,62)(50,64)Polycyclic presentation:
2^8|a^2=g,c^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #138936, #138938, #138944, … |
| maximal normal subgroup of | #138936, #138938, #138944, … |
| maximal quotient of | #220182, #220184, #220188, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15448);