Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| 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 | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(10,13)(11,15)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,4,5,3)(2,7,6,8)(10,13)(14,16)(18,19)(20,28)(25,27)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=e,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=deh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23285, #6383323, #6383328, … |
| maximal normal subgroup of | C2×#23285, #6383323, #6383328, … |
| maximal quotient of | #356449, #356793, #356827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16273);