Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
783 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 783 conjugacy classes of subgroups — too many to draw. Only its 151 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(9,10,14,17)(11,15,19,22)(12,21,16,13)(18,24,23,20);(1,2,4,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,20)(14,19)(16,23)(17,22)(21,24);(6,8)(9,12,13,10,14,16,21,17)(11,18,20,15,19,23,24,22);(1,3,4,2)(11,19)(15,22)(18,23)(20,24)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=f,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139513, #139518, #152754, … |
| maximal normal subgroup of | #139513, #139518, #152754, … |
| maximal quotient of | #297312, #297314, #297324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18721);