Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11858 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | SD16 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
195 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,15,21,14,20)(11,22,24,38,26,32,25,36)(12,27,29,18,31,37,30,17)(16,28,33,23,35,39,34,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,16,21,35)(12,28,31,39)(13,25,14,24)(17,36,18,38)(19,34,20,33)(22,27,32,37)(23,29,40,30);(3,8)(5,7)(9,12,13,29,15,31,14,30)(10,17,20,37,21,18,19,27)(11,23,24,39,26,40,25,28)(16,32,34,38,35,22,33,36)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=gh,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,c^b=cfh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130243, #130253, #130724, … |
| maximal normal subgroup of | #130243, #130253, #130724, … |
| maximal quotient of | #122747, #122749, #122751, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4884);