Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | Aut(G) order 2580480 |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| extraspecial | 2+1+6 |
| Heisenberg | H(7,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C26 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
2996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15);(2,6)(7,13)(8,11)(14,16);(3,9)(7,13)(10,15)(14,16);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)Polycyclic presentation:
2^7|b^a=bg,f^a=fg,e^b=eg,d^c=dg65 conjugacy classes — view the full character table.
| derived subgroup of | H(7,𝔽2)⋊C7, C2×H(7,𝔽2)⋊C7, #63103⋊C7, … |
| Fitting subgroup of | #87536, #87547, H(7,𝔽2)⋊C5, … |
| maximal subgroup of | #32894, #32896, #60388, … |
| maximal normal subgroup of | #32894, #32896, #60388, … |
| maximal quotient of | #62652, #62654, #62842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2326);