Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| 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 |
| prime radical | #3551 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
846 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,24)(22,27)(26,34)(28,36)(29,37)(30,35)(31,40)(32,41)(33,39)(38,44)(42,46)(43,47)(45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23)(20,31)(21,30)(22,29)(24,35)(25,40)(26,38)(27,37)(28,32)(33,45)(34,44)(36,41)(39,48)(42,47)(43,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,24)(21,34)(23,35)(25,39)(26,27)(30,44)(33,46)(36,47)(37,38)(40,48)(41,42)(43,45);(3,11)(6,9)(10,16)(13,15)(17,20)(18,25)(19,28)(21,33)(22,32)(23,36)(24,39)(26,42)(27,41)(29,31)(30,43)(34,46)(35,47)(37,40)(38,48)(44,45);(2,8)(6,9)(7,14)(13,15)(19,29)(20,32)(23,37)(25,41)(30,44)(33,46)(35,38)(39,42)Polycyclic presentation:
2^8|c^2=f,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh,e^d=eh49 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#61801, #10577994, #10578020, … |
| maximal normal subgroup of | C2×#61801, #10577994, #10578020, … |
| maximal quotient of | #6386190, #6387935, #6387937, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54789);