Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2410 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
647 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 647 conjugacy classes of subgroups — too many to draw. Only its 133 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)(3,13,17,8)(4,18)(5,23,25,10)(6,11)(7,12)(9,21)(14,39,42,29)(15,30)(16,37,27,31)(19,46,45,32)(20,47,49,33)(22,38)(24,51,54,35)(26,36)(28,34)(40,43)(41,61)(44,56)(48,64)(50,60,53,58)(52,59)(55,62)(57,63);(1,3,6,17)(2,8,11,13)(4,19,21,45)(5,15,25,43)(7,27,26,16)(9,32,18,46)(10,30,23,40)(12,37,36,31)(14,28,42,22)(20,48,49,63)(24,52,54,44)(29,38,39,34)(33,57,47,64)(35,59,51,56)(41,53,62,50)(55,60,61,58);(1,4)(2,9)(3,14)(5,20)(6,21)(7,28)(8,29)(10,33)(11,18)(12,38)(13,39)(15,41)(16,45)(17,42)(19,27)(22,26)(23,47)(24,53)(25,49)(30,55)(31,46)(32,37)(34,36)(35,60)(40,61)(43,62)(44,63)(48,52)(50,54)(51,58)(56,64)(57,59);(1,5)(2,10)(3,15)(4,20)(6,25)(7,24)(8,30)(9,33)(11,23)(12,35)(13,40)(14,41)(16,44)(17,43)(18,47)(19,48)(21,49)(22,50)(26,54)(27,52)(28,53)(29,55)(31,56)(32,57)(34,58)(36,51)(37,59)(38,60)(39,61)(42,62)(45,63)(46,64)Polycyclic presentation:
2^8|a^2=fh,b^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501111, #501112, #501115, … |
| maximal normal subgroup of | #501111, #501112, #501115, … |
| maximal quotient of | #267374, #267377, #267380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17399);