Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) = C28×D4 (order 2048) · Inn(G) = C22×D4 · Out(G) = C26 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
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 | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10,14,19,15,18,13,27)(11,16,22,31,23,30,21,37)(12,24,25,17,26,33,28,32)(20,34,35,29,36,40,38,39);(1,2)(3,7)(4,8)(5,6)(9,11,14,22,15,23,13,21)(10,16,19,31,18,30,27,37)(12,20,25,35,26,36,28,38)(17,29,33,40,32,39,24,34);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,20)(13,28)(14,25)(15,26)(16,29)(18,32)(19,33)(21,38)(22,35)(23,36)(24,27)(30,39)(31,40)(34,37);(1,4,5,3)(2,7,6,8)(10,18)(16,30)(17,32)(19,27)(24,33)(29,39)(31,37)(34,40)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23082, #6382120, #6382122, … |
| maximal normal subgroup of | C2×#23082, #6382120, #6382122, … |
| maximal quotient of | #230486, #230487, #230488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16070);