Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,17,21,18,22,23,27)(15,19,25,20,16,26,24,28);(6,9)(10,11)(13,15,23,24,18,16,17,25)(14,19,27,28,22,26,21,20);(2,4)(13,16,18,15)(14,20,22,28)(17,25,23,24)(19,27,26,21);(5,7,8,12)(6,10,9,11)(13,17,18,23)(14,21,22,27)(15,24,16,25)(19,28,26,20)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=h,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28428, #6463163, #6463172, … |
| maximal normal subgroup of | C2×#28428, #6463163, #6463172, … |
| maximal quotient of | #320475, #320491, #320502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21416);