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:
(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,18)(19,23)(20,22)(21,28)(24,29)(25,31)(26,27)(30,32)(33,34,39,42)(35,40,36,37)(38,46,47,41)(43,45,44,48);(2,6)(3,9)(8,12)(11,15)(18,20)(23,27)(24,28)(30,31)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,48,46,45);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,48,46,45);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,18)(19,23,26,27)(21,28,29,24)(25,30,32,31)(33,37,39,40)(34,36,42,35)(38,45,47,48)(41,44,46,43);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,21,22,29)(18,24,20,28)(19,25,26,32)(23,30,27,31)(33,38,39,47)(34,41,42,46)(35,43,36,44)(37,45,40,48)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=gh,d^2=fg,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 | #7228611, #7228615, #7228641, … |
| maximal normal subgroup of | #7228611, #7228615, #7228641, … |
| maximal quotient of | #582267, #582453, #582731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46145);