Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
293 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 151 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 — 4 generators on 48 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,26,23,27,21,36)(19,24,30,38,31,39,29,45)(20,32,35,40,33,46,34,25)(28,41,44,47,42,48,43,37);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,30,23,31,21,29)(18,24,26,38,27,39,36,45)(20,28,35,44,33,42,34,43)(25,37,32,41,40,47,46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,34,23,33,22,35)(18,25,26,32,27,40,36,46)(19,28,29,43,31,42,30,44)(24,37,38,41,39,47,45,48);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(20,33)(25,40)(28,42)(32,46)(34,35)(37,47)(41,48)(43,44)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137895, #210415, #210419, … |
| maximal normal subgroup of | #137895, #210415, #210419, … |
| maximal quotient of | #219256, #219257, #219258, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15245);