Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | Aut(G) = C22×#1202 (order 4096) · Inn(G) = C22×C22⋊C4 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
495 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,14,7)(4,8)(5,9)(6,10)(12,16,13,15)(17,18)(19,28)(20,25)(21,26)(22,31)(23,27)(24,29)(30,32);(1,3,6,14)(2,7,10,11)(4,12,5,13)(8,15,9,16)(17,19,23,20)(18,24,27,32)(21,29,22,30)(25,31,28,26);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(17,20,23,19)(18,25,27,28)(21,29,22,30)(24,31,32,26);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(17,21,23,22)(18,26,27,31)(19,29,20,30)(24,25,32,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=g,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #220536, #220540, #220544, … |
| maximal normal subgroup of | #220536, #220540, #220544, … |
| maximal quotient of | #163733, #163739, #163744, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10598);