Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(6,7)(10,11)(13,14)(15,23)(16,19)(17,21)(18,27)(20,25)(22,26)(24,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,16)(14,19,21,23)(18,24,26,25)(20,22,28,27);(2,4)(5,7,9,6)(8,10,12,11)(13,16,17,15)(14,20,21,28)(18,24,26,25)(19,22,23,27);(5,8,9,12)(6,10,7,11)(15,16)(19,23)(20,28)(24,25)Polycyclic presentation:
2^8|b^2=h,c^2=e,d^2=eh,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,d^b=dh,d^c=de,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29702, #6435434, #6435441, … |
| maximal normal subgroup of | C2×#29702, #6435434, #6435441, … |
| maximal quotient of | #328991, #328994, #328997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22690);