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 | |
647 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 647 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)(3,13)(4,18,21,9)(5,10)(6,25)(7,12)(8,16)(11,28)(14,41,44,39)(15,40)(17,37)(19,48,52,32)(20,33,55,49)(22,56)(23,38)(24,36)(26,31)(27,35)(29,43,47,53)(30,46)(34,45)(42,58,62,61)(50,57,63,64)(51,59,54,60);(1,3,7,17)(2,8,12,31)(4,14,21,44)(5,22,24,46)(6,26,28,16)(9,29,18,47)(10,34,36,40)(11,37,25,13)(15,27,45,23)(19,50,52,63)(20,53,55,43)(30,38,56,35)(32,58,48,61)(33,41,49,39)(42,54,62,51)(57,60,64,59);(1,4)(2,9)(3,14)(5,19)(6,20)(7,21)(8,29)(10,32)(11,33)(12,18)(13,39)(15,42)(16,43)(17,44)(22,50)(23,51)(24,52)(25,49)(26,53)(27,54)(28,55)(30,57)(31,47)(34,58)(35,59)(36,48)(37,41)(38,60)(40,61)(45,62)(46,63)(56,64);(1,5,7,24)(2,10,12,36)(3,15,17,45)(4,19,21,52)(6,27,28,23)(8,30,31,56)(9,32,18,48)(11,38,25,35)(13,40,37,34)(14,42,44,62)(16,46,26,22)(20,54,55,51)(29,57,47,64)(33,60,49,59)(39,61,41,58)(43,63,53,50)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=h,e^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 | #138941, #138943, #138949, … |
| maximal normal subgroup of | #138941, #138943, #138949, … |
| maximal quotient of | #220184, #220185, #220189, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15445);