Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29592 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 459 conjugacy classes of subgroups — too many to draw. Only its 123 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)(7,10)(8,9)(11,12)(14,18)(19,27)(20,25)(21,28);(6,9)(10,12)(13,14,16,18)(15,21,22,28)(17,25,24,20)(19,26,27,23);(2,4)(7,11)(10,12)(13,15)(14,19)(16,22)(17,26)(18,27)(20,28)(21,25)(23,24);(5,7,8,11)(6,10,9,12)(15,22)(19,27)(21,28)(23,26)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=eh,g^2=h,b^a=be,c^a=cf,c^b=cg,g^b=gh,d^c=de,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #100712, #100713, #168545, … |
| maximal normal subgroup of | #100712, #100713, #168545, … |
| maximal quotient of | #330179, #330180, #330181, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22789);