Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| socle | C2 | C22×#99 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
610 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 610 conjugacy classes of subgroups — too many to draw. Only its 181 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 — 4 generators on 32 points:
(1,2,6,11,7,12,4,9)(3,13,15,30,14,28,21,32)(5,18,17,29,20,10,22,27)(8,23,26,19,24,31,25,16);(1,3)(2,8)(4,15)(5,19)(6,21)(7,14)(9,25)(10,28)(11,26)(12,24)(13,18)(16,20)(17,23)(22,31)(27,32)(29,30);(3,14)(8,24)(13,28)(15,21)(16,19)(23,31)(25,26)(30,32);(1,4,7,6)(2,9,12,11)(3,15,14,21)(5,17,20,22)(8,25,24,26)(10,27,18,29)(13,30,28,32)(16,31,19,23)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,f^2=h,b^a=be,e^a=eg,g^a=gh,c^b=ch,e^b=eh,f^b=fgh,f^e=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #146655, #146658, #146674, … |
| maximal normal subgroup of | #146655, #146658, #146674, … |
| maximal quotient of | #480003, #480034, #480109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26357);