Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
579 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 579 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) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,20,17,22,16,21)(15,19,25,26,23,27,24,28);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,24)(17,23)(18,25)(20,26)(21,28)(22,27);(2,4)(14,20)(16,18)(19,26)(21,22)(24,25)(27,28);(6,10)(9,11)(15,23)(19,27)(24,25)(26,28)Polycyclic presentation:
2^8|a^2=fh,b^2=eg,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #142546, #142550, C2×#28320, … |
| maximal normal subgroup of | #142546, #142550, C2×#28320, … |
| maximal quotient of | #319677, #319679, #319685, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21308);