Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25140 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
332 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(1,2,5,8)(3,7,4,6)(9,10,12,16,13,17,20,15)(11,14,19,21,18,22,24,23);(1,3,5,4)(2,6,8,7)(9,11,12,19,13,18,20,24)(10,14,16,21,17,22,15,23);(10,15)(11,18)(12,20)(14,21)(16,17)(22,23);(1,4,5,3)(2,7,8,6)Polycyclic presentation:
2^8|a^2=ef,b^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh43 conjugacy classes — view the full character table.
| maximal subgroup of | #285120, #285122, #285135, … |
| maximal normal subgroup of | #285120, #285122, #285135, … |
| maximal quotient of | #290075, #290082, #290087, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18477);