Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | Aut(G) = C22×#7622880 (order 2048) · Inn(G) = C2×#120 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C4×#120 |
| prime radical | #3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
641 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,19)(3,14,17,9,18,28,23,27)(4,10,20,5,11,21,24,22)(15,16,30,32,26,29,31,25);(1,3)(2,9)(4,15)(5,16)(6,23)(7,17)(8,18)(10,25)(11,26)(12,28)(13,27)(14,19)(20,30)(21,32)(22,29)(24,31);(1,4)(2,10)(3,15)(5,12)(6,24)(7,20)(8,11)(9,25)(13,21)(14,29)(16,28)(17,30)(18,26)(19,22)(23,31)(27,32);(1,5,7,21,8,22,6,10)(2,11,12,24,13,4,19,20)(3,16,17,32,18,29,23,25)(9,26,28,31,27,15,14,30)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=eh,f^b=fh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #495768, #495770, #495779, … |
| maximal normal subgroup of | #495768, #495770, #495779, … |
| maximal quotient of | #208744, #208745, #208746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14466);