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) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
295 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 295 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)(13,14,19,24)(15,25,16,27)(17,23,18,22)(20,28,21,26);(1,2)(3,4)(7,11)(9,12)(13,15)(14,20)(16,19)(17,28)(18,26)(21,24)(22,27)(23,25);(2,4)(5,6)(7,12)(8,10)(9,11)(13,16,19,15)(14,21,24,20)(17,28,18,26)(22,27,23,25)Polycyclic presentation:
2^8|a^2=g,c^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×#10082, #233630, #233639, … |
| maximal normal subgroup of | C2×#10082, #233630, #233639, … |
| maximal quotient of | #102508, #102518, #102528, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3070);