Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 427 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:
(5,6)(7,11)(8,9)(10,12)(13,14,20,25)(15,22,26,28)(16,21,17,27)(18,24,19,23);(1,2)(3,4)(6,9)(10,11)(13,15,19,17,20,26,18,16)(14,21,24,28,25,27,23,22);(2,4)(13,16,19,15,20,17,18,26)(14,22,24,21,25,28,23,27);(5,7,8,12)(6,10,9,11)(13,17,19,26,20,16,18,15)(14,22,24,21,25,28,23,27)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=eh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #503914, #504432, #504558, … |
| maximal normal subgroup of | #503914, #504432, #504558, … |
| maximal quotient of | #307612, #307618, #307624, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19251);