Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#399 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 165 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:
(1,2,5,9,6,10,12,20)(3,11,14,22,15,26,19,7)(4,16,18,30,17,8,23,25)(13,27,29,24,28,32,31,21);(1,3,12,14,6,15,5,19)(2,7,20,11,10,22,9,26)(4,13,23,29,17,28,18,31)(8,24,30,32,16,21,25,27);(1,4)(2,8)(3,13)(5,18)(6,17)(7,21)(9,25)(10,16)(11,27)(12,23)(14,29)(15,28)(19,31)(20,30)(22,24)(26,32);(4,17)(8,16)(13,28)(18,23)(21,24)(25,30)(27,32)(29,31)Polycyclic presentation:
2^8|a^2=e,b^2=ef,e^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #23723⋊C3 |
| maximal subgroup of | #142396, #142397, #396522, … |
| maximal normal subgroup of | #142396, #142397, #396522, … |
| maximal quotient of | #243839, #243840, #243841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16711);