Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2048 |
| automorphism | Aut(G) = C23×#72236189 (order 8192) · Inn(G) = #394 · Out(G) = C27 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
299 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 299 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(5,6,11,7,9,13,10,15)(8,14,19,20,17,16,18,12);(5,7,10,13,9,15,11,6)(8,16,18,12,17,14,19,20);(5,8,9,17)(6,12)(7,14)(10,18,11,19)(13,20)(15,16)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,d^b=dh,g^b=gh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11274, #358283, C4×#2048, … |
| maximal normal subgroup of | C2×#11274, #358283, C4×#2048, … |
| maximal quotient of | #120667, #120668, #120718, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4262);