Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
545 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 545 conjugacy classes of subgroups — too many to draw. Only its 175 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) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(3,7)(6,8)(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,23,24,22)(26,27)(28,32);(2,5)(6,8)(10,11)(14,15)(19,24)(22,23)(25,26,29,27)(28,30,32,31);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,23)(21,24)(25,27,29,26)(28,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,23,24,22)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^8|a^2=ef,b^2=efgh,c^2=fg,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | #144499, #144507, #145839, … |
| maximal normal subgroup of | #144499, #144507, #145839, … |
| maximal quotient of | #155765, #155824, #156067, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9991);