Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18224 |
| automorphism | Aut(G) = C2×#72286737 (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 | C22×C4 | C24 |
| prime radical | #3535 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 469 conjugacy classes of subgroups — too many to draw. Only its 113 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)(4,11)(7,13)(8,12)(10,15)(14,16)(17,18,21,24)(19,22,26,29)(20,23,27,30)(25,31,32,28);(1,2,5,8)(3,7,9,14)(4,12,11,6)(10,16,15,13)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,15)(12,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(3,9)(7,14)(10,15)(13,16)Polycyclic presentation:
2^8|a^2=h,b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #346503, #346504, #346507, … |
| maximal normal subgroup of | #346503, #346504, #346507, … |
| maximal quotient of | #166300, #166316, #166355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11222);