Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3068 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
717 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,11,14,9,23)(7,12,18,27,17,28,24,26)(8,19,22,29,20,33,21,13)(16,30,32,25,31,36,34,35);(1,2,3,4)(5,7,9,18,11,17,10,24)(6,12,15,26,14,28,23,27)(8,16,21,32,20,31,22,34)(13,25,19,30,29,35,33,36);(5,8,9,21,11,20,10,22)(6,13,15,19,14,29,23,33)(7,16,18,32,17,31,24,34)(12,25,26,30,28,35,27,36);(8,20)(13,29)(16,31)(19,33)(21,22)(25,35)(30,36)(32,34)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=f,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #366638, #366639, #366731, … |
| maximal normal subgroup of | #366638, #366639, #366731, … |
| maximal quotient of | #204431, #204437, #204442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14238);