Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | Aut(G) = C2×#72236189 (order 2048) · Inn(G) = #2668 · Out(G) = C24 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2668 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3020 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 105 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)(3,15)(4,21)(5,11)(6,13)(7,12)(8,14)(9,18)(10,23)(16,34)(17,43)(19,33)(20,32)(22,51)(24,41)(25,56)(26,39)(27,38)(28,40)(29,36)(30,46)(31,47)(35,53)(37,48)(42,45)(44,57)(49,58)(50,54)(52,60)(55,61)(59,63)(62,64);(1,3,8,20)(2,9,14,33)(4,16,24,46)(5,25,28,47)(6,19,7,18)(10,34,36,30)(11,37,40,43)(12,32,13,15)(17,27,48,26)(21,42,41,58)(22,52,54,64)(23,49,29,45)(31,39,56,38)(35,60,61,62)(44,55,63,53)(50,57,51,59);(1,4)(2,10)(3,16)(5,22)(6,23)(7,29)(8,24)(9,30)(11,35)(12,21)(13,41)(14,36)(15,42)(17,44)(18,45)(19,49)(20,46)(25,52)(26,53)(27,55)(28,54)(31,57)(32,58)(33,34)(37,62)(38,51)(39,50)(40,61)(43,60)(47,64)(48,63)(56,59);(1,5,8,28)(2,11,14,40)(3,17,20,48)(4,22,24,54)(6,27,7,26)(9,31,33,56)(10,35,36,61)(12,39,13,38)(15,43,32,37)(16,44,46,63)(18,47,19,25)(21,50,41,51)(23,55,29,53)(30,57,34,59)(42,60,58,62)(45,64,49,52)Polycyclic presentation:
2^8|b^2=h,d^2=h,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #6632111, #6632113, #6632118, … |
| maximal normal subgroup of | #6632111, #6632113, #6632118, … |
| maximal quotient of | #332010, #332012, #332013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23428);