Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(6,7)(10,12)(13,15)(18,22)(23,24);(2,3)(4,8)(9,10,14,12)(11,13,16,15);(1,2,5,3)(4,6,8,7)(9,11,14,16)(10,15,12,13)(17,18)(19,23)(20,24)(21,22);(1,3,5,2)(4,7,8,6)(9,12,14,10)(11,13,16,15)(17,19,21,20)(18,23,22,24);(1,4,5,8)(2,6,3,7)(9,13)(10,11)(12,16)(14,15)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45918, #7702409, #7702511, … |
| maximal normal subgroup of | C2×#45918, #7702409, #7702511, … |
| maximal quotient of | #557312, #557324, #557515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38906);