Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29902 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2622 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2622 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 101 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)(3,8)(4,10)(5,11)(6,9)(7,12)(13,23)(14,21)(15,22)(16,20)(17,26)(18,24)(19,25)(27,32)(28,30)(29,31);(1,3,6,13,5,16,7,15)(2,8,12,22,11,20,9,23)(4,14,18,27,17,28,19,29)(10,21,25,31,26,30,24,32);(2,9)(3,13)(4,17)(6,7)(8,20)(10,25)(11,12)(14,27)(15,16)(21,30)(24,26)(28,29);(1,4)(2,10)(3,14)(5,17)(6,18)(7,19)(8,21)(9,24)(11,26)(12,25)(13,27)(15,29)(16,28)(20,30)(22,31)(23,32)Polycyclic presentation:
2^8|b^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,c^b=cg,e^b=eh,d^c=de,f^c=fh,g^c=gh,g^d=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #500250, #500252, #500256, … |
| maximal normal subgroup of | #500250, #500252, #500256, … |
| maximal quotient of | #330923, #330927, #330929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22890);