Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12767 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1790 |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
364 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,7,11,21,28)(3,9,14,26,4,15,17,29)(5,10,19,27,18,23,20,30)(8,22,24,16,25,31,32,13);(1,3,7,4)(2,8,11,25)(5,13,18,16)(6,17,21,14)(9,23,15,10)(12,24,28,32)(19,31,20,22)(26,30,29,27);(1,4,7,3)(2,9,11,15)(5,13,18,16)(6,14,21,17)(8,23,25,10)(12,29,28,26)(19,31,20,22)(24,27,32,30)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,e^b=egh,f^b=fg,d^c=dg,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152571, #152574, #152590, … |
| maximal normal subgroup of | #152571, #152574, #152590, … |
| maximal quotient of | #129577, #129580, #129619, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5759);