Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3498 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1067 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 479 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,16)(11,18)(13,14)(17,23)(19,24)(21,22);(2,5)(6,8)(9,10,13,16,15,18,14,11)(12,17,21,23,20,24,22,19);(1,3,4,7)(2,6,5,8)(9,11,13,10,15,16,14,18)(12,19,21,17,20,23,22,24);(9,12)(10,17)(11,19)(13,21)(14,22)(15,20)(16,23)(18,24);(12,20)(17,24)(19,23)(21,22)Polycyclic presentation:
2^8|b^2=gh,c^2=f,f^2=h,g^2=h,b^a=bf,c^a=cg,f^a=fh,g^a=gh,c^b=ch,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60626⋊C3 |
| maximal subgroup of | #129974, #152074, #249813, … |
| maximal normal subgroup of | #129974, #152074, #249813, … |
| maximal quotient of | #6371057, #6371701, #6371958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53614);