Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 393216 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
751 conjugacy classes of subgroups; only the 399 normal subgroups are shown.
This group has 751 conjugacy classes of subgroups — too many to draw. Only its 399 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(5,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,24)(22,23)(25,26,30,32)(27,33,34,29)(28,31,38,40)(35,39,36,37);(1,2,4,6)(3,5,7,8)(10,14)(11,17)(12,21)(22,24)(27,34)(29,33)(35,36)(37,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,18)(12,19,21,23)(16,22,20,24)(25,27)(26,29)(28,36)(30,34)(31,39)(32,33)(35,38)(37,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,31)(27,35)(29,37)(30,38)(32,40)(33,39)(34,36);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,27)(28,37)(30,33)(31,35)(32,34)(36,40)(38,39)Polycyclic presentation:
2^8|a^2=fh,b^2=fgh,c^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| Fitting subgroup of | #47196⋊C3 |
| maximal subgroup of | #133804, #251869, #251919, … |
| maximal normal subgroup of | #133804, #251869, #251919, … |
| maximal quotient of | #576615, #577219, #577929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40184);