Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
511 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 511 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)(3,12)(4,14)(5,17)(9,22)(10,18)(11,24)(13,27)(15,31)(16,19)(20,29)(23,25)(26,32)(28,30);(1,2)(3,9)(4,15)(5,18)(6,11)(7,10)(8,17)(12,25)(13,28)(14,30)(16,23)(19,31)(20,32)(21,26)(22,27)(24,29);(1,3,7,4)(2,9,10,15)(5,19,17,27)(6,12,21,14)(8,22,18,31)(11,25,26,30)(13,29,16,20)(23,32,28,24);(1,4,7,3)(2,9,10,15)(5,16,17,13)(6,14,21,12)(8,23,18,28)(11,25,26,30)(19,20,27,29)(22,32,31,24)Polycyclic presentation:
2^8|c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,d^b=dg,e^b=eg,f^b=fh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #245617, #245621, #245629, … |
| maximal normal subgroup of | #245617, #245621, #245629, … |
| maximal quotient of | #168917, #168919, #168929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11342);