Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) = #72234537 (order 8192) · Inn(G) = C24 · Out(G) = #6342371 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10,13,14)(11,15,16,12)(18,22)(23,25)(24,28)(31,32);(3,7)(6,8)(11,16)(12,15)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)Polycyclic presentation:
2^7|a^2=eg,b^2=e,c^2=e,d^2=e,b^a=be,c^a=cf,c^b=cg,d^b=def,d^c=de26 conjugacy classes — view the full character table.
| maximal subgroup of | #12201, #12277, C2×#2833, … |
| maximal normal subgroup of | #12201, #12277, C2×#2833, … |
| maximal quotient of | #14626, #14634, #14639, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1565);