Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) = #72303263 (order 2048) · Inn(G) = C2×#99 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C2×M16 | C23 |
| socle | C2 | #1459 |
| prime radical | #532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
182 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 182 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) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,12,8,13,20,32,7,14,24,25,5,11,16,31)(3,15,18,28,17,27,4,10,19,29,22,30,23,9,21,26);(1,3)(2,9)(4,20)(5,23)(6,18)(7,19)(8,17)(10,12)(11,29)(13,15)(14,27)(16,21)(22,24)(25,26)(28,31)(30,32);(1,4,8,22,7,21,5,18)(2,10,13,30,14,26,11,28)(3,16,17,6,19,20,23,24)(9,25,15,31,27,12,29,32)Polycyclic presentation:
2^8|a^2=e,c^2=g,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,f^a=fh,c^b=ch,d^b=dh,e^b=efh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #129595, #129596, #129613, … |
| maximal normal subgroup of | #129595, #129596, #129613, … |
| maximal quotient of | #122378, #122379, #122380, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4474);