Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | Aut(G) = D4×#442 (order 512) · Inn(G) = #447 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8)(3,12)(4,15)(5,6)(9,23)(10,27)(11,22)(13,29)(14,19)(16,20)(17,18)(21,24)(25,32)(26,30);(1,2)(3,9)(4,16)(5,10)(6,20)(7,11)(8,15)(12,24)(13,30)(14,26)(17,22)(18,27)(19,25)(21,28)(23,29)(31,32);(1,3,7,14)(2,9,11,26)(4,13,18,29)(5,19,6,12)(8,21,22,32)(10,25,20,24)(15,28,17,31)(16,30,27,23)Polycyclic presentation:
2^7|c^2=g,d^2=fg,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=eg,f^a=fg,d^b=df,f^b=fg,e^c=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#2193, #33538, #33540, … |
| maximal normal subgroup of | C2×#2193, #33538, #33540, … |
| maximal quotient of | #12109, #12110, #12165, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 925);