Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C23 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
144 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 144 conjugacy classes of subgroups — too many to draw. Only its 61 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) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,8,5,7,13,15)(3,9,10,16,11,6,12,14)(17,18,22,27,20,29,36,26,23,28,39,44,37,25,21,38)(19,30,33,42,35,43,48,46,34,45,47,24,31,41,32,40);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,34)(18,24,28,42)(20,31,37,35)(21,33,36,47)(22,32,39,48)(25,40,29,46)(26,43,38,41)(27,45,44,30);(2,7)(6,9)(8,15)(14,16)(17,20,23,37)(18,25,28,29)(19,31,34,35)(21,22,36,39)(24,40,42,46)(26,27,38,44)(30,43,45,41)(32,47,48,33)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=fgh,g^2=h,b^a=bd,c^a=cf,d^a=df,d^b=dg,e^b=efg,g^b=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131098, #131099, #131101, … |
| maximal normal subgroup of | #131098, #131099, #131101, … |
| maximal quotient of | #124274, #124473, #124702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4961);