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 | #2539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1109 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)(7,11)(8,9)(10,12)(13,14,18,19)(15,20,23,27)(16,21,22,17)(24,25,26,28);(6,9)(10,11)(14,19)(15,23)(17,21)(24,26);(1,2,3,4)(13,15)(14,20)(16,26)(17,25)(18,23)(19,27)(21,28)(22,24);(5,6)(7,10)(8,9)(11,12)(13,16,18,22)(14,21,19,17)(15,24,23,26)(20,25,27,28);(5,7,8,12)(6,10,9,11)(13,17,18,21)(14,22,19,16)(15,25,23,28)(20,24,27,26)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=h,e^2=f,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×#50888, #7706663, #7706686, … |
| maximal normal subgroup of | C2×#50888, #7706663, #7706686, … |
| maximal quotient of | #560447, #560487, #561544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43876);