Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) = C2×#72296209 (order 8192) · Inn(G) = #120 · Out(G) = C2×#2846 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #2819 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
324 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(6,9)(7,11)(10,12)(13,14,18,22)(15,23,25,28)(16,21,17,20)(19,26,27,24);(1,2)(3,4)(7,11)(9,12)(13,15)(14,19)(16,26)(17,24)(18,25)(20,28)(21,23)(22,27);(2,4)(5,6)(7,12)(8,10)(9,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,c^b=cf,d^b=dg,e^b=eh,d^c=dh,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10081, #233630, #233634, … |
| maximal normal subgroup of | C2×#10081, #233630, #233634, … |
| maximal quotient of | #102506, #102516, #102524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3069);