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 | #3487 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
606 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 606 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)(3,13)(4,9)(5,10)(6,20,21,11)(7,12)(8,15,23,19)(14,24)(16,29,30,25)(17,26)(18,27)(22,28,32,31);(1,3,7,5)(2,8,12,23)(4,14,17,18)(6,15,21,19)(9,22,26,32)(10,11,13,20)(16,28,30,31)(24,29,27,25);(1,4,7,17)(2,9,12,26)(3,14,5,18)(6,16,21,30)(8,22,23,32)(10,27,13,24)(11,25,20,29)(15,28,19,31);(1,5,7,3)(2,10,12,13)(4,14,17,18)(6,19,21,15)(8,11,23,20)(9,24,26,27)(16,28,30,31)(22,29,32,25)Polycyclic presentation:
2^8|a^2=f,b^2=h,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 | #33351⋊C3 |
| maximal subgroup of | #493497, #493499, #493501, … |
| maximal normal subgroup of | #493497, #493499, #493501, … |
| maximal quotient of | #480013, #480051, #480098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26339);