Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33359 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×Q8 | C24 |
| prime radical | #3500 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
604 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 604 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:
(1,2,7,12,8,13,4,10)(3,14,15,30,17,25,22,32)(5,11,19,29,6,20,18,28)(9,23,26,21,27,31,24,16);(1,3)(2,9)(4,15)(5,16)(6,21)(7,22)(8,17)(10,24)(11,25)(12,26)(13,27)(14,20)(18,23)(19,31)(28,30)(29,32);(1,4,8,7)(2,10,13,12)(3,15,17,22)(5,19,6,18)(9,24,27,26)(11,29,20,28)(14,30,25,32)(16,31,21,23);(1,5,8,6)(2,11,13,20)(3,16,17,21)(4,18,7,19)(9,25,27,14)(10,28,12,29)(15,23,22,31)(24,30,26,32)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=h,e^2=h,f^2=h,b^a=be,e^a=eg,g^a=gh,e^b=eh,f^b=fgh,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #33371⋊C3 |
| maximal subgroup of | #151826, #152705, #152739, … |
| maximal normal subgroup of | #151826, #152705, #152739, … |
| maximal quotient of | #480015, #480047, #480119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26359);