Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60768 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
850 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,28)(20,32)(21,26)(22,24)(23,29)(25,34)(27,36)(30,38)(31,37)(33,41)(35,42)(39,43)(40,46)(44,48)(45,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,33)(21,29)(22,31)(24,39)(25,41)(26,28)(27,37)(30,34)(32,38)(35,45)(36,43)(40,47)(42,48)(44,46);(3,10)(6,9)(11,16)(13,15)(18,24)(19,29)(22,36)(23,37)(25,40)(26,27)(28,39)(30,33)(32,42)(35,46)(38,47)(41,48);(2,7)(6,9)(8,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,38)(24,40)(26,32)(27,42)(28,41)(29,33)(31,44)(36,46)(37,47)(39,48)(43,45);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(20,34)(25,32)(30,33)(35,46)(38,41)(40,42)(44,45)(47,48)Polycyclic presentation:
2^8|e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dh,g^c=gh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61010, #10575864, #10575897, … |
| maximal normal subgroup of | C2×#61010, #10575864, #10575897, … |
| maximal quotient of | #6374694, #6374695, #6375042, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53998);