Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#99 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| socle | C2 | C22×#99 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
626 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 181 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 — 4 generators on 32 points:
(1,2)(3,11,12,24)(4,8)(5,17,19,9)(6,10)(7,14)(13,25,26,32)(15,28,30,22)(16,23)(18,20)(21,27)(29,31);(1,3)(2,7)(4,13)(5,18)(6,12)(8,21)(9,24)(10,20)(11,17)(14,19)(15,29)(16,26)(22,32)(23,31)(25,28)(27,30);(3,12)(7,20)(11,24)(13,26)(14,18)(21,31)(25,32)(27,29);(1,4,6,16)(2,8,10,23)(3,13,12,26)(5,15,19,30)(7,21,20,31)(9,22,17,28)(11,25,24,32)(14,27,18,29)Polycyclic presentation:
2^8|a^2=f,d^2=h,e^2=h,b^a=be,e^a=eg,g^a=gh,c^b=ch,e^b=eh,f^b=fgh,f^e=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #146653, #146660, #146673, … |
| maximal normal subgroup of | #146653, #146660, #146673, … |
| maximal quotient of | #480003, #480034, #480109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26346);