Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | Aut(G) = D4×#12067 (order 2048) · Inn(G) = #2184 · Out(G) = C2×C8 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| prime radical | #3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 137 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,8,14)(3,15,21,59)(4,22,27,47)(5,28,33,87)(6,34,35,12)(7,36,38,94)(9,39,44,17)(10,26,48,24)(11,49,52,32)(13,53,55,30)(16,60,64,114)(18,58,20,57)(19,69,70,111)(23,74,80,122)(25,46,83,108)(29,54,90,101)(31,92,93,85)(37,50,68,112)(40,97,99,117)(41,65,43,66)(42,88,102,89)(45,103,106,125)(51,91,86,110)(56,67,113,62)(61,71,96,100)(63,118,119,104)(72,109,105,78)(73,81,95,76)(75,115,124,98)(77,84,79,121)(82,116,120,128)(107,126,127,123);(1,3)(2,9)(4,16)(5,29)(6,18)(7,37)(8,21)(10,40)(11,50)(12,41)(13,54)(14,44)(15,28)(17,32)(19,30)(20,35)(22,71)(23,75)(24,62)(25,82)(26,67)(27,64)(31,88)(33,90)(34,43)(36,42)(38,68)(39,49)(45,104)(46,107)(47,100)(48,99)(51,111)(52,112)(53,70)(55,101)(56,84)(57,85)(58,92)(59,87)(60,74)(61,78)(63,76)(65,91)(66,110)(69,86)(72,116)(73,115)(77,123)(79,126)(80,124)(81,119)(83,120)(89,93)(94,102)(95,98)(96,109)(97,103)(105,128)(106,118)(108,127)(113,121)(114,122)(117,125);(1,4,8,27)(2,10,14,48)(3,16,21,64)(5,23,33,80)(6,26,35,24)(7,25,38,83)(9,40,44,99)(11,45,52,106)(12,47,34,22)(13,46,55,108)(15,56,59,113)(17,61,39,96)(18,67,20,62)(19,63,70,119)(28,84,87,121)(29,75,90,124)(30,76,53,81)(31,79,93,77)(32,78,49,109)(36,95,94,73)(37,82,68,120)(41,100,43,71)(42,98,102,115)(50,104,112,118)(51,105,86,72)(54,107,101,127)(57,114,58,60)(65,97,66,117)(69,116,111,128)(74,85,122,92)(88,126,89,123)(91,103,110,125)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=fg,e^2=h,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fg,g^a=gh,d^b=df,f^b=fg,g^b=gh,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13672, #512848, #512850, … |
| maximal normal subgroup of | C2×#13672, #512848, #512850, … |
| maximal quotient of | #151180, #151181, #151292, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6660);