Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle, prime radical | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,25,21,24,26,19)(20,28,30,27,29,23,31,32);(1,2)(3,7)(4,8)(5,6)(9,11,13,12)(10,14,15,16)(17,19,26,24,21,25,22,18)(20,27,31,28,29,32,30,23);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,21,29)(18,23,24,28)(19,27,25,32)(22,30,26,31);(1,4,5,3)(2,7,6,8)(20,29)(23,28)(27,32)(30,31)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^b=de,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23259, #6383179, #6383182, … |
| maximal normal subgroup of | C2×#23259, #6383179, #6383182, … |
| maximal quotient of | #356395, #356911, #356928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16247);