Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
486 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 486 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) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(4,15)(8,21)(9,19)(10,16)(12,14)(13,28)(17,30)(20,25)(22,32)(23,29)(24,31)(26,27);(1,2,5,10)(3,8,12,22)(4,16,15,7)(6,9,18,24)(11,25,14,23)(13,29,27,20)(17,31,30,19)(21,26,32,28);(3,12)(4,17)(7,19)(8,22)(11,14)(13,26)(15,30)(16,31)(20,21)(23,25)(27,28)(29,32);(1,3,6,14)(2,8,9,23)(4,13,17,28)(5,12,18,11)(7,20,19,32)(10,22,24,25)(15,27,30,26)(16,29,31,21)Polycyclic presentation:
2^8|b^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #244548, #244549, #244559, … |
| maximal normal subgroup of | #244548, #244549, #244559, … |
| maximal quotient of | #168940, #168953, #168996, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11447);