Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,10)(11,12)(13,14,19,15,18,21,17,23)(16,20,24,22,25,28,26,27);(5,6)(7,11)(8,12)(9,10)(13,15,19,21,18,23,17,14)(16,22,24,28,25,27,26,20);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,18,25)(14,20,21,28)(15,22,23,27)(17,26,19,24);(5,8,9,7)(6,11,10,12)(13,17,18,19)(14,15,21,23)(16,24,25,26)(20,27,28,22)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23211, #6382833, #6382837, … |
| maximal normal subgroup of | C2×#23211, #6382833, #6382837, … |
| maximal quotient of | #232026, #232027, #232028, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16199);