Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13309 |
| automorphism | Aut(G) = C2×#72231730 (order 4096) · Inn(G) = #469 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #469 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #2054 |
| prime radical | #3414 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 53 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)(3,4)(5,18)(6,26)(7,28)(8,13)(9,14)(10,20)(11,25)(12,24)(15,30)(16,32)(17,22)(19,23)(21,29)(27,31);(1,3)(2,9)(4,18)(5,23)(6,10)(7,29)(8,17)(11,21)(12,19)(13,30)(14,25)(15,27)(16,24)(20,28)(22,26)(31,32);(1,4,5,19,24,32,27,30,8,22,6,20,7,21,25,9)(2,10,11,17,28,15,26,16,13,23,31,3,12,14,18,29)Polycyclic presentation:
2^8|c^2=d,d^2=gh,e^2=gh,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gh,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gh,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93846, #93847, #151297, … |
| maximal normal subgroup of | #93846, #93847, #151297, … |
| maximal quotient of | #145365, #145366, #145387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6432);