Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1339 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)(5,6,11,16)(7,12,18,26)(8,21,23,30)(9,15,10,14)(13,24,29,22)(17,31,32,36)(19,28,20,27)(25,34,35,33);(1,2,3,4)(5,7)(6,12)(8,17)(9,19)(10,20)(11,18)(13,25)(14,27)(15,28)(16,26)(21,31)(22,33)(23,32)(24,34)(29,35)(30,36);(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,21)(15,30)(16,29)(18,32)(19,33)(20,34)(26,35)(27,31)(28,36);(7,18)(12,26)(17,32)(19,20)(25,35)(27,28)(31,36)(33,34);(5,9,11,10)(6,14,16,15)(7,19,18,20)(8,22,23,24)(12,27,26,28)(13,21,29,30)(17,33,32,34)(25,31,35,36)Polycyclic presentation:
2^8|a^2=h,b^2=f,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=eh,g^a=gh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #368856, #368858, #368886, … |
| maximal normal subgroup of | #368856, #368858, #368886, … |
| maximal quotient of | #6355325, #6355326, #6355327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53415);