Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23918 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
499 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,21)(15,22,23,28)(16,20,17,25)(19,26,27,24);(1,2,3,4)(5,7,8,12)(6,9,10,11)(15,23)(19,27)(22,28)(24,26);(2,4)(13,15)(14,19)(16,24)(17,26)(18,23)(20,28)(21,27)(22,25);(6,10)(9,11)(13,16,18,17)(14,20,21,25)(15,24,23,26)(19,28,27,22)Polycyclic presentation:
2^8|a^2=h,b^2=egh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#28125, #6459396, #6459407, … |
| maximal normal subgroup of | C2×#28125, #6459396, #6459407, … |
| maximal quotient of | #316259, #316262, #316265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21113);