Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19713 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
394 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 115 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,27)(20,23)(21,31)(22,26)(24,29)(25,28)(30,32);(1,2,7,13,5,9,4,6)(3,8,14,16,10,15,11,12)(17,19,28,30,22,29,21,20)(18,23,31,24,26,32,25,27);(1,3,5,10)(2,8,9,15)(4,11,7,14)(6,12,13,16)(17,20,21,29,22,30,28,19)(18,24,25,23,26,27,31,32);(3,10)(8,15)(11,14)(12,16)(19,29)(20,30)(23,32)(24,27)Polycyclic presentation:
2^8|b^2=eg,c^2=fgh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dh,d^c=dg40 conjugacy classes — view the full character table.
| maximal subgroup of | #143166, C2×#19729, #6357368, … |
| maximal normal subgroup of | #143166, C2×#19729, #6357368, … |
| maximal quotient of | #191526, #206266, #206272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12717);