Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32950 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2847 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 649 conjugacy classes of subgroups — too many to draw. Only its 145 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:
(2,7)(3,12)(5,8)(10,20)(11,21)(13,28)(14,15)(16,17)(19,23)(22,31)(25,26)(27,29);(1,2,8,21,6,11,5,7)(3,9,12,23,15,24,14,19)(4,10,17,25,18,26,16,20)(13,22,27,32,29,31,28,30);(1,3)(2,9)(4,13)(5,14)(6,15)(7,19)(8,12)(10,22)(11,24)(16,28)(17,27)(18,29)(20,30)(21,23)(25,32)(26,31);(1,4)(2,10)(3,13)(5,16)(6,18)(7,20)(8,17)(9,22)(11,26)(12,27)(14,28)(15,29)(19,30)(21,25)(23,32)(24,31)Polycyclic presentation:
2^8|b^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,f^c=fh,g^c=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #144225, #144226, #144240, … |
| maximal normal subgroup of | #144225, #144226, #144240, … |
| maximal quotient of | #347881, #347882, #347883, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25939);