Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
200 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 200 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,13)(11,15)(14,16)(17,18)(19,23,28,37)(20,30)(21,34)(22,25)(24,32)(26,31)(27,39,41,48)(29,43,40,36)(33,44)(35,46,47,42)(38,45);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19)(18,23,25,37)(20,27)(21,29)(22,28)(24,35,38,47)(26,36,34,43)(30,39,44,48)(31,40)(32,42)(33,41)(45,46);(2,7)(6,14)(8,12)(9,15)(17,20,31,45,22,33,21,32)(18,24,34,44,25,38,26,30)(19,27,40,46,28,41,29,42)(23,35,43,48,37,47,36,39)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=eg,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9757, #205535, #205538, … |
| maximal normal subgroup of | C2×#9757, #205535, #205538, … |
| maximal quotient of | #102033, #102034, #102047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2745);