Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3335 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,11)(4,9)(5,7)(6,10)(8,15)(12,24)(13,20)(14,25)(16,23)(17,19)(18,27)(21,30)(22,28)(26,32)(29,31);(2,7)(3,12)(4,16)(9,22)(10,20)(14,29)(15,27)(17,28)(19,23)(21,25)(26,30)(31,32);(1,3,13,27)(2,8,20,24)(4,14,28,29)(5,12,6,15)(7,18,10,11)(9,21,22,32)(16,30,17,26)(19,31,23,25);(2,7)(8,18)(9,19)(10,20)(11,24)(21,31)(22,23)(25,32);(1,4,5,17)(2,9,7,19)(3,14,12,26)(6,16,13,28)(8,21,18,31)(10,23,20,22)(11,25,24,32)(15,30,27,29);(2,10)(7,20)(8,11)(9,23)(18,24)(19,22)(21,25)(31,32)Polycyclic presentation:
2^8|c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| Fitting subgroup of | #62826⋊C3 |
| maximal subgroup of | #185272, #185278, #185284, … |
| maximal normal subgroup of | #185272, #185278, #185284, … |
| maximal quotient of | #7626213, #7626226, #7626248, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55814);