Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | Aut(G) = C2×#72231746 (order 2048) · Inn(G) = #410 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #410 |
| commutator | C22×C4 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#41 | C23 |
| socle | C2 | #1820 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 53 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)(3,12,15,10)(4,16)(5,9)(6,11)(7,14,21,28)(8,13,23,27)(17,20)(18,25)(19,24)(22,26,31,32)(29,30);(1,3)(2,7,11,21)(4,13)(5,14)(6,15)(8,20,23,30)(9,12,24,10)(16,22,25,31)(17,26)(18,27)(19,28)(29,32);(1,4,5,17,6,18,19,29)(2,8,9,22,11,23,24,31)(3,13,14,26,15,27,28,32)(7,20,12,25,21,30,10,16)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=d,d^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,f^b=fh,f^c=fg,f^e=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94192, #94193, #150602, … |
| maximal normal subgroup of | #94192, #94193, #150602, … |
| maximal quotient of | #135191, #135194, #135221, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5852);