Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator, socle, prime radical | C23 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(19,25)(22,28)(26,31)(29,32)Polycyclic presentation:
2^7|a^2=f,b^2=de,c^2=d,f^2=g,b^a=bd,c^a=ce,c^b=cg44 conjugacy classes — view the full character table.
| Fitting subgroup of | #1840⋊C3 |
| maximal subgroup of | C2×#1840, #20773, #20774, … |
| maximal normal subgroup of | C2×#1840, #20773, #20774, … |
| maximal quotient of | #8802, #10796, #11298, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 572);