Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 101 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 — 4 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27)(19,23,28,38)(20,24,33,41)(21,34,32,26)(25,42,40,31)(29,39,43,35)(30,47,44,37)(36,45,48,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,39)(25,36)(26,37)(27,38)(31,46)(32,44)(33,43)(34,47)(35,41)(40,48)(42,45);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,28)(20,31)(21,32)(23,35)(25,26)(27,41)(29,45)(33,42)(34,40)(36,47)(37,48)(38,39)(43,46);(2,8)(6,9)(7,16)(14,15)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ceh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #139019, #139023, #139139, … |
| maximal normal subgroup of | #139019, #139023, #139139, … |
| maximal quotient of | #496432, #496483, #496738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26790);