Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
691 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 691 conjugacy classes of subgroups — too many to draw. Only its 147 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 — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14)(11,15,18,20)(12,16,19,21)(17,23,24,22);(1,3,4,8)(2,5,6,7)(10,14)(15,20)(16,21)(22,23);(9,11)(10,15)(12,17)(13,18)(14,20)(16,22)(19,24)(21,23);(9,12)(10,16)(11,17)(13,19)(14,21)(15,22)(18,24)(20,23)Polycyclic presentation:
2^8|a^2=e,b^2=eh,b^a=be,c^a=cf,d^a=dg,g^c=gh,f^d=fh49 conjugacy classes — view the full character table.
| Fitting subgroup of | #32977⋊C3 |
| maximal subgroup of | #500491, #500492, #500493, … |
| maximal normal subgroup of | #500491, #500492, #500493, … |
| maximal quotient of | #349226, #349227, #349228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25965);