Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | Aut(G) = C2×#72218858 (order 46080) · Inn(G) = C24 · Out(G) = C22×S6 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C24 |
| commutator, socle | C2 | C24×C4 |
| Frattini, cosocle | C4 | C25 |
| prime radical | #585 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ 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:
(1,2,5,9,6,10,16,22)(3,7,13,19,12,18,25,29)(4,8,15,21,14,20,26,30)(11,17,24,28,23,27,31,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,17)(9,19)(10,18)(14,23)(15,24)(16,25)(20,27)(21,28)(22,29)(26,31)(30,32);(1,4)(2,8)(3,11)(5,15)(6,14)(7,17)(9,21)(10,20)(12,23)(13,24)(16,26)(18,27)(19,28)(22,30)(25,31)(29,32);(4,14)(8,20)(11,23)(15,26)(17,27)(21,30)(24,31)(28,32);(3,12)(7,18)(11,23)(13,25)(17,27)(19,29)(24,31)(28,32)Polycyclic presentation:
2^7|a^2=f,f^2=g,e^b=eg,d^c=dg68 conjugacy classes — view the full character table.
| derived subgroup of | #72341774, #72341798, #72341844, … |
| Fitting subgroup of | #85629, #85666, #3573⋊C5, … |
| maximal subgroup of | #12684, #12685, #21477, … |
| maximal normal subgroup of | #12684, #12685, #21477, … |
| maximal quotient of | #60058, #60065, #60100, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2305);