Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| automorphism | Aut(G) order 262144 |
| 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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
929 conjugacy classes of subgroups; only the 403 normal subgroups are shown.
This group has 929 conjugacy classes of subgroups — too many to draw. Only its 403 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10,13,14)(11,15,16,12)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(1,3,5,8)(2,6,7,4)(10,14)(12,15)(18,22)(19,26)(20,29)(30,31);(9,11)(10,12)(13,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,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.
| Frattini subgroup of | #72267638 |
| maximal subgroup of | #130051, #133427, #133813, … |
| maximal normal subgroup of | #130051, #133427, #133813, … |
| maximal quotient of | #576600, #576621, #577210, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39825);