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) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#555 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
570 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 570 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:
(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,20)(15,18,22,26)(16,19,23,27)(21,25,24,28);(6,9)(10,11)(15,21)(16,23)(18,25)(19,27)(22,24)(26,28);(2,4)(13,15,17,22)(14,18,20,26)(16,24,23,21)(19,28,27,25);(5,7,8,12)(6,10,9,11)(15,22)(18,26)(21,24)(25,28)Polycyclic presentation:
2^8|a^2=h,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,d^c=dh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250620, #250624, #250635, … |
| maximal normal subgroup of | #250620, #250624, #250635, … |
| maximal quotient of | #258161, #258162, #258163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17121);