Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9930 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | #3192 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
305 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 305 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(2,6)(3,4)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,21)(19,23)(22,24);(1,2)(3,8)(4,6)(5,7)(9,11,13,18)(10,14,16,21)(12,17,19,24)(15,22,23,20);(1,2)(3,6)(4,8)(5,7)(9,12)(10,15)(11,17)(13,19)(14,20)(16,23)(18,24)(21,22)Polycyclic presentation:
2^8|b^2=h,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=eg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9932, #234820, #234821, … |
| maximal normal subgroup of | C2×#9932, #234820, #234821, … |
| maximal quotient of | #102723, #102724, #102725, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2920);