Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | #2397 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 457 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(2,6)(3,4)(11,16)(13,15);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^8|a^2=g,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cef,d^b=de,d^c=dh64 conjugacy classes — view the full character table.
| Fitting subgroup of | #23358⋊C3 |
| maximal subgroup of | #136302, #137285, #139970, … |
| maximal normal subgroup of | #136302, #137285, #139970, … |
| maximal quotient of | #232169, #232170, #232171, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16346);