Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9431 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #386 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3153 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
288 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,8,5,3,4,7)(9,10,13,17,14,18,12,15)(11,19,21,16,22,23,20,24);(1,3,5,2)(4,7,6,8)(10,15,18,17)(16,19,24,23);(4,6)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,19)(17,23)(18,24)Polycyclic presentation:
2^8|a^2=f,b^2=dfh,d^2=gh,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152179, #152483, C2×#9441, … |
| maximal normal subgroup of | #152179, #152483, C2×#9441, … |
| maximal quotient of | #100113, #100115, #100119, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2429);