Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | #3486 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
595 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 595 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)(3,8)(4,9)(5,17,18,7)(6,10)(11,21)(12,26,27,19)(13,22)(14,23)(15,29,30,20)(16,24)(25,32,28,31);(2,7)(8,19)(9,20)(10,17)(21,31)(22,26)(23,32)(24,29);(1,3,6,13)(2,8,10,22)(4,14,16,11)(5,12,18,27)(7,19,17,26)(9,23,24,21)(15,28,30,25)(20,32,29,31);(1,4,6,16)(2,9,10,24)(3,11,13,14)(5,15,18,30)(7,20,17,29)(8,21,22,23)(12,25,27,28)(19,31,26,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,b^a=be,e^a=eg,g^a=gh,f^b=fg,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #33349⋊C3 |
| maximal subgroup of | #493493, #493495, #493505, … |
| maximal normal subgroup of | #493493, #493495, #493505, … |
| maximal quotient of | #480011, #480051, #480099, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26337);