Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 32768 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,4)(9,10)(11,20)(12,23)(13,17)(14,18)(15,19)(16,21)(22,24);(5,6)(7,8)(9,11,14,21,15,12,13,22)(10,16,17,20,19,24,18,23);(2,4)(6,8)(9,12)(10,16)(11,15)(13,22)(14,21)(17,23)(18,20)(19,24);(9,13,15,14)(10,17,19,18)(11,21,12,22)(16,23,24,20)Polycyclic presentation:
2^8|a^2=ef,b^2=e,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #395719, #395720, #395721, … |
| maximal normal subgroup of | #395719, #395720, #395721, … |
| maximal quotient of | #242443, #242446, #242449, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16604);