Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,18)(12,16)(13,23)(14,17)(15,21)(19,20)(22,24);(9,11,13,21,14,22,12,20)(10,15,16,18,17,19,23,24);(1,2,3,4)(9,10)(11,19)(12,23)(13,16)(14,17)(15,22)(18,20)(21,24);(5,7,8,6)(9,12,14,13)(10,16,17,23)(11,20,22,21)(15,18,19,24)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=f,d^2=gh,e^2=h,b^a=be,c^a=cf,e^a=eh,c^b=ch,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #219490, #219504, #219507, … |
| maximal normal subgroup of | #219490, #219504, #219507, … |
| maximal quotient of | #212968, #212970, #212972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14692);