Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1569 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)(5,6,10,14)(7,15,16,28)(8,12,19,26)(9,21,22,13)(11,23,24,18)(17,29,30,36)(20,33,32,27)(25,34,35,31);(5,7,9,18,10,16,22,23)(6,11,13,15,14,24,21,28)(8,17,20,31,19,30,32,34)(12,25,27,29,26,35,33,36);(1,2,3,4)(7,16)(11,24)(15,28)(17,30)(18,23)(25,35)(29,36)(31,34);(5,8)(6,12)(7,17)(9,20)(10,19)(11,25)(13,27)(14,26)(15,29)(16,30)(18,31)(21,33)(22,32)(23,34)(24,35)(28,36);(8,19)(12,26)(17,30)(20,32)(25,35)(27,33)(29,36)(31,34)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=g,f^2=h,b^a=bf,c^a=cg,f^a=fh,c^b=ch,e^d=eh58 conjugacy classes — view the full character table.
| Fitting subgroup of | #60651⋊C3 |
| maximal subgroup of | #152052, #249801, #249803, … |
| maximal normal subgroup of | #152052, #249801, #249803, … |
| maximal quotient of | #6357223, #6357225, #6357227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53639);