Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) = #72238017 (order 2048) · Inn(G) = C2×#99 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#99 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C2×Q8 | C23 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,11,7,12,4,9)(3,13,16,31,17,28,14,30)(5,19,18,29,21,10,22,27)(8,23,24,20,25,32,26,15);(1,3,7,17)(2,8,12,25)(4,16,6,14)(5,20,21,15)(9,26,11,24)(10,28,19,13)(18,23,22,32)(27,30,29,31);(1,4,7,6)(2,9,12,11)(3,14,17,16)(5,18,21,22)(8,24,25,26)(10,27,19,29)(13,30,28,31)(15,23,20,32)Polycyclic presentation:
2^7|a^2=e,b^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,d^a=df,f^a=fg,c^b=cg,d^b=dg,e^b=efg,e^d=eg23 conjugacy classes — view the full character table.
| maximal subgroup of | #12712, #12713, #12721, … |
| maximal normal subgroup of | #12712, #12713, #12721, … |
| maximal quotient of | #11434, #11450, #11481, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 867);