Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| automorphism | Aut(G) = #72317013 (order 4096) · Inn(G) = #2021 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2021 |
| commutator, Frattini, cosocle | C2×#41 | C23 |
| prime radical | #585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
335 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 335 conjugacy classes of subgroups — too many to draw. Only its 47 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:
(1,2,8,13)(3,14,18,40)(4,19,24,44)(5,25,29,11)(6,30,7,12)(9,26,36,16)(10,23,39,22)(15,47,49,63)(17,52,31,46)(20,45,53,64)(21,56,27,54)(28,42,32,43)(33,57,62,55)(34,50,35,51)(37,48,61,60)(38,58,41,59);(1,3)(2,9)(4,20)(5,26)(6,17)(7,31)(8,18)(10,37)(11,40)(12,34)(13,36)(14,25)(15,23)(16,29)(19,33)(21,57)(22,49)(24,53)(27,55)(28,51)(30,35)(32,50)(38,47)(39,61)(41,63)(42,52)(43,46)(44,62)(45,56)(48,58)(54,64)(59,60);(1,4,8,24)(2,10,13,39)(3,15,18,49)(5,27,29,21)(6,23,7,22)(9,33,36,62)(11,41,25,38)(12,44,30,19)(14,45,40,64)(16,48,26,60)(17,53,31,20)(28,58,32,59)(34,37,35,61)(42,54,43,56)(46,63,52,47)(50,57,51,55)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,f^b=fh,g^b=gh,d^c=dg,e^c=eh,f^c=fh,f^d=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #151156, #151157, #151320, … |
| maximal normal subgroup of | #151156, #151157, #151320, … |
| maximal quotient of | #142231, #142278, #142327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6350);