Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| 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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1579 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)(7,15,17,29)(8,12,19,26)(9,21,22,13)(11,23,25,18)(16,28,30,36)(20,33,32,27)(24,34,35,31);(5,7,9,18,10,17,22,23)(6,11,13,15,14,25,21,29)(8,16,20,31,19,30,32,34)(12,24,27,28,26,35,33,36);(1,2,3,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,28)(17,30)(18,31)(21,33)(22,32)(23,34)(25,35)(29,36);(8,19)(12,26)(16,30)(20,32)(24,35)(27,33)(28,36)(31,34);(7,17)(11,25)(15,29)(16,30)(18,23)(24,35)(28,36)(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,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #152048, #152050, #246929, … |
| maximal normal subgroup of | #152048, #152050, #246929, … |
| maximal quotient of | #6357173, #6357174, #6357176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53633);