Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3460 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
505 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 505 conjugacy classes of subgroups — too many to draw. Only its 119 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:
(2,3)(6,7)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23);(1,2)(3,5)(4,6)(7,8)(10,14)(13,20)(17,24)(19,22);(1,3,5,2)(4,6,8,7)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17);(1,4,5,8)(2,6,3,7)(11,18)(15,23)(17,24)(19,22)Polycyclic presentation:
2^8|c^2=e,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gh,c^b=cg,f^b=fh,d^c=de,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #487489, #487492, #487501, … |
| maximal normal subgroup of | #487489, #487492, #487501, … |
| maximal quotient of | #328793, #328803, #328804, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22695);