Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
216 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,5,6,7)(9,10,12,16,13,14,11,15)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,2)(3,7)(4,6)(5,8)(10,14)(11,12)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(9,11,13,12)(10,15,14,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^8|a^2=e,c^2=f,d^2=g,e^2=h,b^a=bd,d^a=df,c^b=ch,d^b=dg,e^b=efg58 conjugacy classes — view the full character table.
| maximal subgroup of | #97360, #97362, #97363, … |
| maximal normal subgroup of | #97360, #97362, #97363, … |
| maximal quotient of | #96268, #96746, #96747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 644);