Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| socle | C22 | C23×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,12)(8,20)(9,14)(10,23)(11,15)(13,21)(16,30)(17,26)(18,27)(19,33)(22,28)(24,29)(25,32)(31,36)(34,35);(1,2,3,4)(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,25)(14,27)(15,26)(20,30)(21,32)(22,31)(23,33)(24,34)(28,36)(29,35);(5,8)(6,13)(7,16)(9,21)(10,24)(11,22)(12,25)(14,28)(15,29)(17,31)(18,32)(19,34)(20,23)(26,35)(27,36)(30,33);(7,17)(12,26)(16,31)(18,19)(25,35)(27,33)(30,36)(32,34);(5,9,11,10)(6,14,15,23)(7,18,17,19)(8,21,22,24)(12,27,26,33)(13,28,29,20)(16,32,31,34)(25,36,35,30)Polycyclic presentation:
2^8|b^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,g^a=gh,d^b=dh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #368846, #368847, #368867, … |
| maximal normal subgroup of | #368846, #368847, #368867, … |
| maximal quotient of | #6354461, #6354462, #6354463, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53410);