Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
478 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 478 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) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24);(1,2,4,6)(3,8,7,5)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24);(3,7)(5,8)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(2,6)(5,8)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)Polycyclic presentation:
2^8|b^2=g,d^2=gh,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh37 conjugacy classes — view the full character table.
| maximal subgroup of | #244556, #244557, #244562, … |
| maximal normal subgroup of | #244556, #244557, #244562, … |
| maximal quotient of | #166327, #166341, #166383, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12032);