Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
695 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 695 conjugacy classes of subgroups — too many to draw. Only its 179 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) = 256 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(2,4)(13,14,18,22)(15,23,26,27)(16,21,17,19)(20,24,28,25);(6,8)(13,15,18,26)(14,20,22,28)(16,25,17,24)(19,23,21,27)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,d^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=dh,f^b=fh,d^c=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #92847, #137381, #137382, … |
| maximal normal subgroup of | #92847, #137381, #137382, … |
| maximal quotient of | #111220, #111312, #111317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4012);