Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| 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 | C2×#562 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,9)(10,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,28)(26,27);(6,9)(7,11)(14,17)(15,21)(20,24)(22,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,21)(14,19,17,23)(16,22,25,27)(20,26,24,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,25)(14,20,17,24)(15,22,21,27)(19,26,23,28);(5,7)(6,10)(8,11)(9,12)(13,17,18,14)(15,23,21,19)(16,24,25,20)(22,28,27,26)Polycyclic presentation:
2^8|c^2=gh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45790, #7702476, #7702503, … |
| maximal normal subgroup of | C2×#45790, #7702476, #7702503, … |
| maximal quotient of | #557069, #557071, #557379, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38778);