Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 | |
660 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)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,22,24,28)(16,21,18,25)(20,26,27,23);(5,7,8,12)(6,9,10,11)(14,19)(20,27)(21,25)(22,28);(13,15)(14,20)(16,23)(17,24)(18,26)(19,27)(21,28)(22,25);(2,4)(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22)Polycyclic presentation:
2^8|a^2=e,b^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #6470446, #6470453, #6470454, … |
| maximal normal subgroup of | #6470446, #6470453, #6470454, … |
| maximal quotient of | #349204, #349207, #349222, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25949);