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 | #2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1036 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)(13,14,18,19)(15,16,20,23)(17,25,26,21)(22,28,27,24);(5,6,8,9)(7,11,12,10)(14,19)(16,23)(21,25)(24,28);(1,2)(3,4)(6,9)(10,11)(13,15,18,20)(14,16,19,23)(17,22,26,27)(21,24,25,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,27)(22,25)(26,28);(5,7,8,12)(6,10,9,11)(13,17)(14,21)(15,22)(16,24)(18,26)(19,25)(20,27)(23,28)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,e^2=h,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×#51892, #7707929, #7707967, … |
| maximal normal subgroup of | C2×#51892, #7707929, #7707967, … |
| maximal quotient of | #560342, #560377, #561447, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 44880);