Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) = #72245616 (order 6144) · Inn(G) = C22×D4 · Out(G) = C23×S4 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D4 |
| commutator, Frattini, cosocle | C4 | C25 |
| socle | C2 | C23×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,12)(4,9)(5,10)(6,19)(7,11)(8,14)(13,26)(15,22)(16,24)(17,25)(18,29)(20,23)(21,28)(27,32)(30,31);(1,3)(2,8)(4,13)(5,14)(6,20)(7,15)(9,21)(10,22)(11,23)(12,19)(16,27)(17,28)(18,30)(24,31)(25,32)(26,29);(1,4)(2,9)(3,13)(5,17)(6,18)(7,16)(8,21)(10,25)(11,24)(12,26)(14,28)(15,27)(19,29)(20,30)(22,32)(23,31);(4,16)(9,24)(13,27)(17,18)(21,31)(25,29)(26,32)(28,30);(1,5,7,6)(2,10,11,19)(3,14,15,20)(4,17,16,18)(8,22,23,12)(9,25,24,29)(13,28,27,30)(21,32,31,26)Polycyclic presentation:
2^7|e^2=g,f^2=g,b^a=bf,f^a=fg,f^b=fg,d^c=dg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #3584⋊C3, #11711520, #11711525, … |
| maximal subgroup of | #12745, #12746, #12747, … |
| maximal normal subgroup of | #12745, #12746, #12747, … |
| maximal quotient of | #60086, #60108, #60161, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2316);