Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | Aut(G) = C2×#72236721 (order 4096) · Inn(G) = #2410 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
663 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 663 conjugacy classes of subgroups — too many to draw. Only its 133 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:
(1,2,5,11,7,8,6,12)(3,13,17,22,16,24,14,9)(4,18,19,26,20,10,21,25)(15,27,28,23,29,32,30,31);(2,8)(3,14)(5,6)(9,13)(10,18)(15,28)(16,17)(19,21)(22,24)(23,27)(29,30)(31,32);(1,3)(2,9)(4,15)(5,14)(6,17)(7,16)(8,22)(10,23)(11,24)(12,13)(18,31)(19,30)(20,29)(21,28)(25,27)(26,32);(1,4)(2,10)(3,15)(5,21)(6,19)(7,20)(8,18)(9,23)(11,26)(12,25)(13,27)(14,28)(16,29)(17,30)(22,31)(24,32)Polycyclic presentation:
2^8|a^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gh,e^c=eh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #501145, #501148, #501151, … |
| maximal normal subgroup of | #501145, #501148, #501151, … |
| maximal quotient of | #267467, #267471, #267476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17408);