Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,24)(17,20)(18,22)(21,23)(25,28)(26,27);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,17)(14,19,22,20)(16,25,26,23)(21,24,28,27);(2,4)(5,7,8,11)(6,9,10,12)(14,20)(15,17)(19,22)(21,24)(23,25)(27,28);(6,10)(9,12)(13,16,18,26)(14,21,22,28)(15,23,17,25)(19,27,20,24)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6633081, #6633085, #6633091, … |
| maximal normal subgroup of | #6633081, #6633085, #6633091, … |
| maximal quotient of | #336904, #336906, #336912, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23670);