Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,4)(3,7,8,6)(18,22)(23,24);(2,4)(6,7)(10,13)(11,15);(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,24)(21,22);(1,3)(2,6)(4,7)(5,8)(17,19,21,20)(18,23,22,24);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^8|a^2=f,c^2=h,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#38309, #7635560, #7635639, … |
| maximal normal subgroup of | C2×#38309, #7635560, #7635639, … |
| maximal quotient of | #521741, #522350, #522862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 31297);