Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,9)(7,12)(10,15)(14,16)(18,22)(19,25)(24,29)(26,31)(33,34)(35,41)(36,38)(37,44)(39,42)(40,45)(43,46)(47,48);(2,6)(4,11)(7,12)(10,15)(18,22)(23,28)(24,29)(30,32)(34,36)(35,42)(40,44)(43,48);(1,2)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,39)(36,41)(37,43)(38,42)(40,46)(44,47)(45,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36,38,34)(35,41,42,39)(37,44,45,40)(43,47,48,46);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,37,38,45)(34,40,36,44)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^8|c^2=fg,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #53388⋊C3 |
| maximal subgroup of | C2×#53388, #8105943, #8105944, … |
| maximal normal subgroup of | C2×#53388, #8105943, #8105944, … |
| maximal quotient of | #606852, #607492, #607493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46376);