Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24378 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,17,21,18,22,16,20)(15,23,24,27,25,28,26,19);(1,2)(3,4)(5,7,9,8)(6,10,12,11);(2,4)(13,15,18,25)(14,19,22,27)(16,24,17,26)(20,23,21,28);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^8|a^2=ef,b^2=e,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24840, #6445608, #6445626, … |
| maximal normal subgroup of | C2×#24840, #6445608, #6445626, … |
| maximal quotient of | #279248, #279254, #279260, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17828);