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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1090 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)(7,11)(8,9)(10,12)(13,14,18,16)(15,19,22,23)(17,20,26,24)(21,27,25,28);(6,9)(10,11)(13,15,18,22)(14,19,16,23)(17,25,26,21)(20,28,24,27);(1,2,3,4)(13,15)(14,19)(16,23)(17,21)(18,22)(20,27)(24,28)(25,26);(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,22,23)(17,24,26,20)(21,27,25,28);(5,7,8,12)(6,10,9,11)(13,17)(14,20)(15,21)(16,24)(18,26)(19,27)(22,25)(23,28)Polycyclic presentation:
2^8|a^2=h,b^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×#51202, #7706863, #7706882, … |
| maximal normal subgroup of | C2×#51202, #7706863, #7706882, … |
| maximal quotient of | #560446, #560475, #561542, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44190);