Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) = C23×#612424 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,17,14,12,20,16)(11,15,18,23,19,21,24,22);(1,3)(2,5)(4,8)(6,7)(9,11,13,18,14,19,20,24)(10,15,17,23,12,21,16,22);(3,8)(5,7)(9,12)(10,14)(11,15)(13,16)(17,20)(18,23)(19,21)(22,24);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=g,b^2=g,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #137920, #209049, #209051, … |
| maximal normal subgroup of | #137920, #209049, #209051, … |
| maximal quotient of | #217600, #217601, #217602, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15049);