Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23444 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
589 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,16)(13,17)(14,18)(19,29)(20,27)(21,28)(22,30)(23,31)(24,32)(25,33)(26,35)(34,39)(36,38)(37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,29)(17,27)(18,28)(22,34)(23,35)(24,36)(25,37)(26,31)(30,39)(32,38)(33,40);(9,12,14,23)(10,16,18,31)(11,19,21,35)(13,24,25,22)(15,26,28,29)(17,32,33,30)(20,36,37,34)(27,39,40,38);(9,13,14,25)(10,17,18,33)(11,20,21,37)(12,22,23,24)(15,27,28,40)(16,30,31,32)(19,34,35,36)(26,38,29,39)Polycyclic presentation:
2^8|a^2=e,b^2=e,c^2=h,d^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #23470⋊C3 |
| maximal subgroup of | #142321, #142323, #142449, … |
| maximal normal subgroup of | #142321, #142323, #142449, … |
| maximal quotient of | #237768, #237769, #237770, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16458);