Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 263 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 263 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,14,10,13,22,27)(7,11,17,25,16,24,32,35)(8,18,19,28,20,12,21,26)(15,29,31,36,30,23,33,34);(1,2,3,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,23)(13,24)(14,25)(18,29)(19,31)(20,30)(21,33)(22,32)(26,34)(27,35)(28,36);(5,8,10,20)(6,12,13,18)(7,15,16,30)(9,21,22,19)(11,23,24,29)(14,28,27,26)(17,33,32,31)(25,36,35,34);(7,16)(11,24)(15,30)(17,32)(23,29)(25,35)(31,33)(34,36)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138315, #138363, #138365, … |
| maximal normal subgroup of | #138315, #138363, #138365, … |
| maximal quotient of | #188274, #188279, #188287, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13627);