Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,4)(3,9,11,7)(5,12,13,8)(10,14,16,15)(17,18,23,27)(19,28,29,24)(20,25,33,39)(21,22,26,36)(30,41,43,37)(31,42,38,32)(34,35,40,46)(44,48,47,45);(1,3,6,11)(2,7,4,9)(5,10,13,16)(8,14,12,15)(19,29)(20,33)(24,28)(25,39)(31,38)(32,42)(34,40)(35,46);(1,4,6,2)(3,7,11,9)(5,12,13,8)(10,14,16,15)(17,19,23,29)(18,24,27,28)(20,30,33,43)(21,31,26,38)(22,32,36,42)(25,37,39,41)(34,44,40,47)(35,45,46,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(29,43)(31,44)(32,45)(36,46)(38,47)(42,48);(17,21)(18,22)(19,31)(20,34)(23,26)(24,32)(25,35)(27,36)(28,42)(29,38)(30,44)(33,40)(37,45)(39,46)(41,48)(43,47);(17,22)(18,26)(19,32)(20,35)(21,27)(23,36)(24,38)(25,40)(28,31)(29,42)(30,45)(33,46)(34,39)(37,47)(41,44)(43,48)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| Fitting subgroup of | #63067⋊C5 |
| maximal subgroup of | #347447, #347467, #347494, … |
| maximal normal subgroup of | #347447, #347467, #347494, … |
| maximal quotient of | #7627421, #7627423, #7627535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56055);