Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9930 |
| automorphism | Aut(G) = C22×#72231762 (order 4096) · Inn(G) = #392 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #392 |
| commutator, Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
263 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 263 conjugacy classes of subgroups — too many to draw. Only its 51 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 — 3 generators on 32 points:
(1,2,7,12)(3,13,4,15)(5,11,6,10)(8,16,9,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32);(1,3)(2,8)(4,7)(5,16)(6,14)(9,12)(10,15)(11,13)(17,19,21,26)(18,22,24,29)(20,25,27,32)(23,30,31,28);(1,4,7,3)(2,9,12,8)(5,16,6,14)(10,15,11,13)(17,20)(18,23)(19,25)(21,27)(22,28)(24,31)(26,32)(29,30)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=g,d^2=g,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=eg31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9937, #234820, #234822, … |
| maximal normal subgroup of | C2×#9937, #234820, #234822, … |
| maximal quotient of | #102728, #102730, #102734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2925);