Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(7,𝔽2) |
| automorphism | Aut(G) order 3317760 |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| extraspecial | 2−1+6 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C26 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
2898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,8)(4,9)(5,17)(6,7)(10,18)(11,21)(12,26)(13,19)(14,23)(15,29)(16,20)(22,27)(24,30)(25,32)(28,31);(2,7)(8,19)(9,20)(10,17)(21,23)(22,26)(24,29)(31,32);(1,3,6,13)(2,8,7,19)(4,14,16,11)(5,12,18,27)(9,23,20,21)(10,22,17,26)(15,28,30,25)(24,32,29,31);(1,4,6,16)(2,9,7,20)(3,11,13,14)(5,15,18,30)(8,21,19,23)(10,24,17,29)(12,25,27,28)(22,31,26,32);(1,2)(3,8)(4,9)(5,10)(6,7)(11,21)(12,22)(13,19)(14,23)(15,24)(16,20)(17,18)(25,31)(26,27)(28,32)(29,30);(1,5)(2,10)(3,12)(4,15)(6,18)(7,17)(8,22)(9,24)(11,25)(13,27)(14,28)(16,30)(19,26)(20,29)(21,31)(23,32)Polycyclic presentation:
2^7|c^2=g,d^2=g,b^a=bg,f^a=fg,e^b=eg,d^c=dg65 conjugacy classes — view the full character table.
| derived subgroup of | #87551, #11710348, #11710353, … |
| Fitting subgroup of | #87537, #87548, #87551, … |
| maximal subgroup of | #32895, #32897, #60390, … |
| maximal normal subgroup of | #32895, #32897, #60390, … |
| maximal quotient of | #62653, #62654, #62845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2327);