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 | |
480 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)(13,14,17,22,19,24,18,23)(15,20,26,28,16,27,25,21);(6,9)(10,11)(13,15,18,25,19,16,17,26)(14,20,23,21,24,27,22,28);(2,4)(13,16,19,15)(14,21,24,28)(17,26,18,25)(20,22,27,23);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,24,23)(15,25,16,26)(20,21,27,28)Polycyclic presentation:
2^8|a^2=fh,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×#28436, #6463163, #6463172, … |
| maximal normal subgroup of | C2×#28436, #6463163, #6463172, … |
| maximal quotient of | #320476, #320490, #320503, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21424);