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 | #2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
549 conjugacy classes of subgroups; only the 131 normal subgroups are shown.
This group has 549 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:
(5,6)(7,11)(8,10)(9,12)(13,14,19,25,18,16,17,21)(15,20,26,28,23,22,24,27);(1,2,3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,22)(17,24)(18,23)(19,26)(21,27)(25,28);(2,4)(13,16,18,14)(15,22,23,20)(17,21,19,25)(24,27,26,28);(6,10)(9,11)(15,23)(20,22)(24,26)(27,28)Polycyclic presentation:
2^8|a^2=fh,b^2=eg,c^2=h,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 | #142545, #142549, C2×#28323, … |
| maximal normal subgroup of | #142545, #142549, C2×#28323, … |
| maximal quotient of | #319681, #319683, #319689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21311);