Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | |
1098 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)(5,6,9,10)(7,8,11,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28);(6,10)(7,11)(14,18)(16,25)(19,21)(22,28);(1,2,3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(5,7)(6,8)(9,11)(10,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(5,8)(6,7)(9,12)(10,11)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)Polycyclic presentation:
2^8|a^2=fh,c^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#48810, #7704182, #7704282, … |
| maximal normal subgroup of | C2×#48810, #7704182, #7704282, … |
| maximal quotient of | #560415, #561517, #561518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 41798);