Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | Aut(G) = C2×#72286730 (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 | #3536 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
477 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 477 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:
(1,2,7,12)(3,13,4,9)(5,11,6,10)(8,15,16,14)(17,18)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)(18,24)(22,29)(23,30)(28,31)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,b^a=be,c^a=cf,d^a=dgh,e^a=eg,e^b=eg,f^b=fh,d^c=dg,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #244527, #244529, #244533, … |
| maximal normal subgroup of | #244527, #244529, #244533, … |
| maximal quotient of | #168918, #168928, #168973, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11445);