Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1217 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 1217 conjugacy classes of subgroups — too many to draw. Only its 463 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(7,11)(10,12)(14,18)(16,23)(19,25)(21,24);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,26)(24,28);(1,2)(3,4)(6,9)(10,12)(13,15,17,22)(14,19,18,25)(16,24,23,21)(20,28,26,27);(5,6)(7,10)(8,9)(11,12)(13,16,17,23)(14,20,18,26)(15,21,22,24)(19,27,25,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,26)(24,28)Polycyclic presentation:
2^8|a^2=g,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #111320, #111470, #111976, … |
| maximal normal subgroup of | #111320, #111470, #111976, … |
| maximal quotient of | #556358, #556386, #556558, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30441);