Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
509 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 509 conjugacy classes of subgroups — too many to draw. Only its 127 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:
(5,6)(7,10)(8,9)(11,12)(13,14,18,21,17,22,16,20)(15,23,26,28,24,19,25,27);(1,2,3,4)(6,9)(10,12)(13,15)(14,19)(16,25)(17,24)(18,26)(20,28)(21,27)(22,23);(2,4)(5,7)(6,10)(8,11)(9,12)(14,20)(15,24)(16,18)(19,27)(21,22)(23,28);(7,11)(10,12)(13,16,17,18)(14,21,22,20)(15,25,24,26)(19,27,23,28)Polycyclic presentation:
2^8|a^2=f,b^2=gh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142805, #142814, #142906, … |
| maximal normal subgroup of | #142805, #142814, #142906, … |
| maximal quotient of | #331118, #331119, #331127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23237);