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) = 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:
(2,8)(3,13)(4,17)(5,14)(9,23)(10,27)(11,20)(12,25)(15,19)(16,18)(21,30)(24,26)(28,31)(29,32);(1,2)(3,9)(4,10)(5,20)(6,12)(7,11)(8,14)(13,24)(15,28)(16,27)(17,26)(18,31)(19,23)(21,32)(22,29)(25,30);(1,3,7,16)(2,9,11,27)(4,14,19,5)(6,13,22,18)(8,23,20,10)(12,24,29,31)(15,30,17,21)(25,26,32,28);(1,4,6,15)(2,10,12,28)(3,14,13,30)(5,18,21,16)(7,19,22,17)(8,24,25,9)(11,23,29,26)(20,31,32,27)Polycyclic presentation:
2^8|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 | #246999, #247001, #247002, … |
| maximal normal subgroup of | #246999, #247001, #247002, … |
| maximal quotient of | #168545, #168546, #168603, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11322);