Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10004 |
| 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 | C2×C42 | C23 |
| prime radical | #532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
267 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 267 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:
(2,7)(3,8)(4,5)(9,10)(11,19)(12,16)(13,22)(14,18)(15,20)(17,23)(21,24);(1,2,6,3)(4,8,5,7)(9,11,14,12)(10,15,18,24)(13,21,23,20)(16,22,19,17);(1,3,6,2)(4,8,5,7)(9,12,14,11)(10,16,18,19)(13,20,23,21)(15,17,24,22)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=g,e^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=egh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10007, #235323, #235324, … |
| maximal normal subgroup of | C2×#10007, #235323, #235324, … |
| maximal quotient of | #102755, #102762, #102767, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2995);