Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,23)(16,25)(17,27)(18,20)(19,24)(21,26)(22,28);(1,2)(3,4)(13,15,18,16)(14,19,20,21)(17,26,28,24)(22,25,27,23);(2,4)(5,7,8,12)(6,9,10,11)(14,20)(19,21)(22,27)(23,25);(13,16,18,15)(14,21,20,19)(17,24,28,26)(22,23,27,25)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=fh,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25965, #6438065, #6438084, … |
| maximal normal subgroup of | C2×#25965, #6438065, #6438084, … |
| maximal quotient of | #305269, #305271, #305296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18953);