Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18316 |
| automorphism | Aut(G) = C22×#72217875 (order 4096) · Inn(G) = C2×#120 · Out(G) = C23×D4 |
| 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 | #3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
449 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 449 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,20)(19,26,29,32)(21,25,22,30)(24,31,27,28);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24)(20,27)(21,28)(22,31)(23,29)(25,32)(26,30);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,18)(19,27,29,24)(21,30,22,25)(26,31,32,28);(2,9)(7,16)(8,13)(10,15)(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)Polycyclic presentation:
2^8|a^2=h,c^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,e^c=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #247000, #247001, #247002, … |
| maximal normal subgroup of | #247000, #247001, #247002, … |
| maximal quotient of | #168557, #168558, #168614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11329);