Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8510 |
| automorphism | Aut(G) = C2×#72233785 (order 3072) · Inn(G) = #379 · Out(G) = C22×A4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3568 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
324 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(1,2,5,12,8,3,14,18)(4,20,23,56,25,46,42,55)(6,28,26,38,31,54,43,49)(7,13,27,10,32,19,44,16)(9,33,35,60,15,45,34,22)(11,39,41,51,17,50,53,30)(21,57,24,40,59,64,61,52)(29,36,62,37,63,47,58,48);(1,3)(2,8)(4,15)(5,10)(6,28)(7,19)(9,25)(11,30)(12,27)(13,32)(14,16)(17,51)(18,44)(20,29)(21,34)(22,40)(23,37)(24,47)(26,53)(31,54)(33,55)(35,59)(36,61)(38,50)(39,49)(41,43)(42,48)(45,56)(46,63)(52,60)(57,62)(58,64);(1,4)(2,9)(3,15)(5,23)(6,29)(7,24)(8,25)(10,37)(11,40)(12,35)(13,36)(14,42)(16,48)(17,52)(18,34)(19,47)(20,28)(21,44)(22,30)(26,62)(27,59)(31,63)(32,61)(33,39)(38,56)(41,64)(43,58)(45,50)(46,54)(49,55)(51,60)(53,57)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg40 conjugacy classes — view the full character table.
| Fitting subgroup of | #8514⋊C3 |
| maximal subgroup of | C2×#8514, #187732, #187733, … |
| maximal normal subgroup of | C2×#8514, #187732, #187733, … |
| maximal quotient of | #97008, #97009, #97014, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1502);