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×#560 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1097 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(15,21)(19,24)(22,27)(26,28);(6,9)(10,11)(14,18)(15,21)(16,23)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28);(5,7,8,12)(6,10,9,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^8|a^2=f,c^2=fh,d^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×#45765, #7702392, #7702502, … |
| maximal normal subgroup of | C2×#45765, #7702392, #7702502, … |
| maximal quotient of | #557030, #557037, #557349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38753);