Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| 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 | #1781 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×#41 | C23 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 73 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 73 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)(3,12,15,25)(4,9,16,7)(5,10)(6,11)(8,14,23,18)(13,28)(17,24,31,21)(19,26)(20,30)(22,32)(27,29);(2,7)(4,16)(8,20)(9,11)(10,21)(12,27)(13,29)(17,31)(22,23)(24,26)(25,28)(30,32);(1,3)(2,8)(4,13)(5,18)(6,15)(7,20)(9,22)(10,25)(11,23)(12,26)(14,19)(16,29)(17,32)(21,28)(24,27)(30,31)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fgh,e^c=eh,f^c=fh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #152025, #152051, #152052, … |
| maximal normal subgroup of | #152025, #152051, #152052, … |
| maximal quotient of | #129236, #129269, #129332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5703);