Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) = #72303099 (order 2048) · Inn(G) = #99 · Out(G) = C26 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #99 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | #1459 |
| prime radical | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
186 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 63 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:
(1,2,6,12,7,11,23,29,8,13,24,31,5,21,15,30)(3,14,17,32,16,9,4,10,18,26,19,27,22,25,20,28);(1,3)(2,9)(4,15)(5,22)(6,19)(7,16)(8,18)(10,12)(11,26)(13,25)(14,21)(17,24)(20,23)(27,29)(28,31)(30,32);(1,4,7,19,8,20,5,17)(2,10,11,27,13,28,21,32)(3,15,16,6,18,23,22,24)(9,12,26,29,25,31,14,30)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,f^a=fh,d^b=dh,e^b=efh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94136, #94137, #94138, … |
| maximal normal subgroup of | #94136, #94137, #94138, … |
| maximal quotient of | #122374, #122375, #122376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4473);