Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10004 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
248 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18,22,26)(19,27,20,24)(21,30,31,25)(23,29,32,28);(1,3,7,4)(2,8,12,9)(5,16,6,14)(10,15,11,13)(17,19,22,20)(18,23,26,32)(21,29,31,28)(24,30,27,25);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,15,11,13)(17,20,22,19)(18,24,26,27)(21,28,31,29)(23,25,32,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10011, #235323, #235325, … |
| maximal normal subgroup of | C2×#10011, #235323, #235325, … |
| maximal quotient of | #102756, #102763, #102768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2999);