Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C22 | C26 |
| commutator | C2 | C25×C4 |
| prime radical | C22×2−1+4 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
5465 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2,3,4)(5,6)(7,12)(8,13)(9,21)(10,11)(14,22)(15,25)(16,30)(17,23)(18,27)(19,33)(20,24)(26,31)(28,34)(29,36)(32,35);(6,11)(12,23)(13,24)(14,21)(25,27)(26,30)(28,33)(35,36);(5,7,10,17)(6,12,11,23)(8,18,20,15)(9,16,22,31)(13,27,24,25)(14,26,21,30)(19,32,34,29)(28,36,33,35);(5,8,10,20)(6,13,11,24)(7,15,17,18)(9,19,22,34)(12,25,23,27)(14,28,21,33)(16,29,31,32)(26,35,30,36);(5,6)(7,12)(8,13)(9,14)(10,11)(15,25)(16,26)(17,23)(18,27)(19,28)(20,24)(21,22)(29,35)(30,31)(32,36)(33,34);(5,9)(6,14)(7,16)(8,19)(10,22)(11,21)(12,26)(13,28)(15,29)(17,31)(18,32)(20,34)(23,30)(24,33)(25,35)(27,36)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg130 conjugacy classes — view the full character table.
| Fitting subgroup of | #62653⋊C3, #62653⋊C5 |
| maximal subgroup of | #392238, #392241, #392278, … |
| maximal normal subgroup of | #392238, #392241, #392278, … |
| maximal quotient of | #7625219, #7625220, #7625222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55641);