Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C25 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
774 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 774 conjugacy classes of subgroups — too many to draw. Only its 153 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)(3,4)(9,10,12,16,13,15,21,23)(11,17,18,22,19,24,20,14);(2,4)(5,6)(7,8);(6,8)(9,11,13,19)(10,14,15,22)(12,20,21,18)(16,24,23,17);(2,4)(10,15)(14,22)(16,23)(17,24)Polycyclic presentation:
2^8|a^2=f,c^2=h,f^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #267787, #267793, #267803, … |
| maximal normal subgroup of | #267787, #267793, #267803, … |
| maximal quotient of | #267213, #267219, #267225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17374);