Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | Aut(G) = #72337551 (order 2048) · Inn(G) = #535 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2535 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
396 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 396 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) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,18)(6,10)(7,13)(8,17)(9,19)(12,25)(14,23)(16,29)(20,28)(21,26)(22,30)(24,27)(31,32);(1,3)(2,7,10,20)(4,12)(5,13)(6,14)(8,21,22,25)(9,11,18,23)(15,24,29,31)(16,26)(17,27)(19,28)(30,32);(4,16)(7,20)(8,22)(11,23)(12,26)(15,29)(17,30)(27,32);(1,4,5,17,6,16,19,30)(2,8,9,15,10,22,18,29)(3,12,13,27,14,26,28,32)(7,21,11,24,20,25,23,31)Polycyclic presentation:
2^8|b^2=g,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #500471, #500472, #500747, … |
| maximal normal subgroup of | #500471, #500472, #500747, … |
| maximal quotient of | #317015, #317016, #317017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21169);