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×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1006 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,22,27)(16,17,20,26)(23,28,25,24);(6,8)(10,11)(15,22)(16,20)(17,26)(19,27);(1,2,3,4)(5,6,9,8)(7,11,12,10)(13,15)(14,19)(16,25)(17,24)(18,22)(20,23)(21,27)(26,28);(5,7,9,12)(6,10,8,11)(13,16,18,20)(14,17,21,26)(15,23,22,25)(19,28,27,24);(5,8,9,6)(7,11,12,10)(13,17)(14,20)(15,24)(16,21)(18,26)(19,23)(22,28)(25,27)Polycyclic presentation:
2^8|a^2=fh,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×#50889, #7706670, #7706688, … |
| maximal normal subgroup of | C2×#50889, #7706670, #7706688, … |
| maximal quotient of | #560435, #560488, #561544, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 43877);