Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 | |
381 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)(7,9)(8,10)(11,12)(15,22)(19,27)(23,28)(24,26);(5,7,9,6)(8,11,12,10)(13,14)(15,23)(16,20)(17,21)(18,25)(19,24)(22,28)(26,27);(2,4)(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,24)(17,22)(18,26)(20,28)(21,27)(23,25);(2,4)(6,7)(10,11)(13,16,17,18)(14,20,21,25)(15,24,22,26)(19,28,27,23)Polycyclic presentation:
2^8|a^2=fh,b^2=e,c^2=e,d^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #6634797, #6634800, #6634810, … |
| maximal normal subgroup of | #6634797, #6634800, #6634810, … |
| maximal quotient of | #344594, #344595, #344598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24129);