Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25798 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2433 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2433 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
715 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 715 conjugacy classes of subgroups — too many to draw. Only its 141 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)(4,16)(5,19)(6,10)(7,14)(8,17)(9,20)(12,25)(13,26)(15,24)(18,29)(21,28)(22,30)(23,27)(31,32);(1,3,5,14,6,15,20,25)(2,7,9,11,10,12,19,24)(4,13,17,27,18,28,30,32)(8,21,16,23,22,26,29,31);(1,2)(3,12,15,7)(4,8)(5,9)(6,10)(11,14,24,25)(13,26,28,21)(16,17)(18,22)(19,20)(23,27,31,32)(29,30);(1,4,5,17,6,18,20,30)(2,8,9,16,10,22,19,29)(3,13,14,27,15,28,25,32)(7,21,11,23,12,26,24,31)Polycyclic presentation:
2^8|b^2=e,c^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gh,c^b=cg,f^b=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #503222, #503223, #503227, … |
| maximal normal subgroup of | #503222, #503223, #503227, … |
| maximal quotient of | #299195, #299197, #299200, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18786);