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 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #532 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,23)(13,18)(14,27)(15,21)(16,25)(19,26)(20,35)(22,30)(24,34)(28,32)(29,39)(31,37)(33,38)(36,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,29)(17,18)(19,30)(23,40)(24,36)(25,37)(26,38)(27,33)(28,39)(31,32)(34,35);(9,12,13,24)(10,16,18,32)(11,20,21,36)(14,28,26,25)(15,29,17,31)(19,34,27,23)(22,39,38,37)(30,35,33,40);(3,8)(5,7)(9,11)(10,17)(12,20)(13,21)(14,22)(15,18)(16,31)(19,33)(23,35)(24,36)(25,37)(26,38)(27,30)(28,39)(29,32)(34,40);(2,6)(5,7)(10,18)(11,21)(16,32)(19,27)(20,36)(22,38)(23,34)(37,39)Polycyclic presentation:
2^8|c^2=h,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×#61681, #10577300, #10577328, … |
| maximal normal subgroup of | C2×#61681, #10577300, #10577328, … |
| maximal quotient of | #6386065, #6387739, #6387741, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54669);