Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | Aut(G) = #72310502 (order 2048) · Inn(G) = #410 · Out(G) = #120 |
| 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 | |
167 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 167 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)(7,14,22,29)(8,13,25,17)(18,21)(19,27)(20,26)(23,32)(24,28,30,31);(1,3)(2,7,11,22)(4,17)(5,14)(6,15)(8,23,25,21)(9,12,26,10)(13,19)(16,30,27,24)(18,31)(20,29)(28,32);(1,4,5,18,6,19,20,32)(2,8,9,24,11,25,26,30)(3,13,14,28,15,17,29,31)(7,21,12,27,22,23,10,16)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,c^b=ch,f^b=fh,f^c=fg,f^e=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #150592, #150593, C2×#12865, … |
| maximal normal subgroup of | #150592, #150593, C2×#12865, … |
| maximal quotient of | #135192, #135193, #135222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5853);