Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | D42 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
314 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 103 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:
(1,2,5,8)(3,9,12,15)(4,13,14,7)(6,10,16,11)(17,18,21,25)(19,22,26,30)(20,23,28,24)(27,31,32,29);(1,3,5,12)(2,6,8,16)(4,10,14,11)(7,15,13,9)(17,19)(18,22)(20,27)(21,26)(23,31)(24,29)(25,30)(28,32);(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(3,10)(4,14)(6,15)(7,13)(9,16)(11,12)Polycyclic presentation:
2^8|a^2=gh,b^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | #153627, C2×#20040, #6359378, … |
| maximal normal subgroup of | #153627, C2×#20040, #6359378, … |
| maximal quotient of | #222352, #222588, #226078, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13028);