Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | Aut(G) = #72305173 (order 4096) · Inn(G) = C2×#102 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | #466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,28,31,44)(20,24)(21,35)(22,27)(23,30,39,48)(25,33)(26,34)(29,42,32,45)(36,38,46,40)(37,41)(43,47);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,23)(20,32)(21,36)(22,31)(24,40)(25,42)(26,38)(27,39)(28,41)(29,34)(30,47)(33,48)(35,44)(37,46)(43,45);(3,9)(6,12)(10,15)(13,16)(17,20,22,34)(18,24,27,26)(19,29,31,32)(21,33,37,47)(23,38,39,40)(25,41,43,35)(28,42,44,45)(30,46,48,36)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,c^b=cgh,d^b=dfh,e^b=eh,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131636, #131637, #131640, … |
| maximal normal subgroup of | #131636, #131637, #131640, … |
| maximal quotient of | #125081, #125083, #125128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4644);