Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | |
1092 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)(7,11)(10,12)(13,14)(15,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,28);(6,9)(10,12)(14,19)(15,20)(21,26)(22,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,19,24)(17,22,25,27)(21,23,26,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,19)(17,23)(18,24)(21,27)(22,26)(25,28);(5,7)(6,10)(8,11)(9,12)(13,17,18,25)(14,21,19,26)(15,22,20,27)(16,23,24,28)Polycyclic presentation:
2^8|c^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#36745, #7634190, #7634221, … |
| maximal normal subgroup of | C2×#36745, #7634190, #7634221, … |
| maximal quotient of | #553772, #553779, #554029, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29733);