Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2308 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3486 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
618 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 618 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,20,24)(3,11,14,25,13,7,18,22)(4,15,16,26,17,8,19,23)(12,21,29,31,28,27,30,32);(1,3)(2,7)(4,12)(5,18)(6,13)(8,21)(9,11)(10,25)(14,20)(15,27)(16,30)(17,28)(19,29)(22,24)(23,32)(26,31);(7,11)(21,27)(22,25)(31,32);(1,4,6,17)(2,8,9,15)(3,12,13,28)(5,19,20,16)(7,21,11,27)(10,26,24,23)(14,30,18,29)(22,32,25,31)Polycyclic presentation:
2^8|a^2=g,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #135532, #139746, #139761, … |
| maximal normal subgroup of | #135532, #139746, #139761, … |
| maximal quotient of | #213722, #213727, #213731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14768);