Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,19)(17,20)(21,28)(22,26)(23,25)(24,27);(6,9)(10,12)(13,15,16,22)(14,18,19,26)(17,21,24,23)(20,28,27,25);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(15,21)(17,24)(18,25)(22,23)(26,28)Polycyclic presentation:
2^7|a^2=eg,b^2=g,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cf,f^b=fg,f^c=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | #7413, C2×#2074, #23627, … |
| maximal normal subgroup of | #7413, C2×#2074, #23627, … |
| maximal quotient of | #10623, #10624, #10626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 806);