Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
419 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,19,24,18,23)(15,20,25,21,16,27,26,28);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,15,17,25,19,16,18,26)(14,20,22,21,24,27,23,28);(6,10)(9,12)(13,16,19,15)(14,21,24,28)(17,25,18,26)(20,23,27,22);(1,3,4,2)Polycyclic presentation:
2^8|a^2=gh,b^2=ef,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #311034, #311042, #311050, … |
| maximal normal subgroup of | #311034, #311042, #311050, … |
| maximal quotient of | #315031, #315032, #315033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20758);