Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10)(11,17,18,24)(12,20,22,15)(13,16)(14,19)(21,23);(9,11,13,18)(10,14,16,23)(12,21,22,19)(15,24,20,17);(1,3,5,4)(2,6,7,8)(11,18)(14,23)(17,24)(19,21)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=g,d^2=h,e^2=g,b^a=bd,c^a=cg,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=efh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #130483, #130485, #130522, … |
| maximal normal subgroup of | #130483, #130485, #130522, … |
| maximal quotient of | #124751, #124767, #124784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4523);