Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
327 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(14,18,20,24)(15,21,23,25);(2,4)(5,7,9,8)(6,10,11,12)(13,14,17,20)(15,22,23,27)(16,24,26,18)(19,25,28,21);(2,4)(5,8,9,7)(6,10,11,12)(13,15,17,23)(14,19,20,28)(16,25,26,21)(18,27,24,22)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=eh,f^c=fh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #92852, #137432, #137441, … |
| maximal normal subgroup of | #92852, #137432, #137441, … |
| maximal quotient of | #121625, #121626, #121627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4364);