Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23900 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | #3485 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
696 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 127 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,20)(19,22)(23,24);(2,5)(6,8)(9,11)(10,14)(12,17)(13,18)(15,21)(16,20)(19,23)(22,24);(1,3,4,7)(2,6,5,8)(14,20)(21,24);(2,5)(6,8)(9,12,13,19)(10,15,16,22)(11,17,18,23)(14,21,20,24)Polycyclic presentation:
2^8|c^2=e,d^2=h,b^a=be,c^a=cf,d^a=de,g^a=gh,c^b=cg,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139765, #139766, #139817, … |
| maximal normal subgroup of | #139765, #139766, #139817, … |
| maximal quotient of | #254581, #254582, #254583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17045);