Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
599 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 599 conjugacy classes of subgroups — too many to draw. Only its 131 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,24)(16,26)(17,23)(18,22)(19,21)(20,25)(27,28);(6,9)(10,11)(13,15,18,25)(14,20,22,24)(16,19,27,17)(21,26,23,28);(1,2,3,4)(13,16,19,25,18,27,17,15)(14,20,23,26,22,24,21,28);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,21,22,23)(15,16,25,27)(20,26,24,28)Polycyclic presentation:
2^8|b^2=h,c^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250782, #250790, #250793, … |
| maximal normal subgroup of | #250782, #250790, #250793, … |
| maximal quotient of | #293989, #293998, #294003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18627);