Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
614 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 614 conjugacy classes of subgroups — too many to draw. Only its 133 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:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,15)(16,24,27,28)(17,21,18,22)(20,26,23,25);(5,7,8,12)(6,9,10,11)(13,15,19,14)(16,24,27,28)(17,22,18,21)(20,26,23,25);(2,4)(13,16,19,27)(14,20,15,23)(17,26,18,25)(21,24,22,28);(6,10)(9,11)(13,17,19,18)(14,21,15,22)(16,25,27,26)(20,28,23,24)Polycyclic presentation:
2^8|a^2=h,b^2=eh,c^2=h,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,c^b=cg,f^b=fh,g^b=gh,d^c=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #139223, #139291, #267957, … |
| maximal normal subgroup of | #139223, #139291, #267957, … |
| maximal quotient of | #251613, #251615, #251617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16971);