Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9331 |
| automorphism | Aut(G) = C2×#72293743 (order 2048) · Inn(G) = #386 · Out(G) = C2×C22⋊C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
180 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 57 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) = 128 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,10,7,12,5,11)(3,13,16,23,17,26,15,8)(4,18,20,9,21,24,19,25)(14,22,29,31,30,32,28,27)(33,34)(35,44,47,55)(36,48)(37,42)(38,52)(39,43)(40,46,56,51)(41,49)(45,60,63,54)(50,57)(53,58)(59,61,64,62);(1,3,7,17)(2,8)(4,14,21,30)(5,15,6,16)(9,22)(10,13)(11,26)(12,23)(18,27)(19,28,20,29)(24,31)(25,32)(33,35,37,51,39,47,38,46)(34,40,42,55,43,56,52,44)(36,45,49,62,50,63,53,61)(41,54,57,64,58,60,48,59);(1,4,7,21)(2,9,12,25)(3,14,17,30)(5,20,6,19)(8,22,23,32)(10,18,11,24)(13,27,26,31)(15,29,16,28)(33,36)(34,41)(35,45)(37,49)(38,53)(39,50)(40,54)(42,57)(43,58)(44,59)(46,61)(47,63)(48,52)(51,62)(55,64)(56,60)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=egh,f^c=fg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9336, #199568, #199570, … |
| maximal normal subgroup of | C2×#9336, #199568, #199570, … |
| maximal quotient of | #99998, #100000, #100006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2324);