Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
609 conjugacy classes of subgroups; only the 399 normal subgroups are shown.
This group has 609 conjugacy classes of subgroups — too many to draw. Only its 399 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) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,10)(8,13)(9,11)(12,15)(14,16)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34,38,41)(35,39,36,43)(37,45,46,40)(42,48,44,47);(2,3)(4,11)(8,10)(12,16)(18,22)(23,25)(24,28)(30,31)(33,35,38,36)(34,39,41,43)(37,42,46,44)(40,47,45,48);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,34)(35,39)(36,43)(37,40)(38,41)(42,47)(44,48)(45,46);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,38,35)(34,39,41,43)(37,44,46,42)(40,47,45,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,37,38,46)(34,40,41,45)(35,42,36,44)(39,47,43,48)Polycyclic presentation:
2^8|a^2=fg,b^2=gh,c^2=h,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #7228627, #7228631, #7228657, … |
| maximal normal subgroup of | #7228627, #7228631, #7228657, … |
| maximal quotient of | #582267, #582454, #582733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46144);