Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2069 |
| automorphism | Aut(G) = #72237738 (order 3072) · Inn(G) = #396 · Out(G) = C22×A4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #396 |
| commutator, Frattini, cosocle | C22×C4 | C23 |
| prime radical | C23 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
98 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,19,6,10,21,9)(3,11,13,7,14,27,31,23)(4,15,17,24,18,8,20,25)(12,28,29,22,16,26,30,32);(1,3,5,13,6,14,21,31)(2,7,9,11,10,23,19,27)(4,16,17,30,18,12,20,29)(8,22,24,28,15,32,25,26);(1,4,6,18)(2,8,10,15)(3,12,14,16)(5,20,21,17)(7,22,23,32)(9,25,19,24)(11,26,27,28)(13,30,31,29)Polycyclic presentation:
2^7|a^2=de,b^2=d,c^2=g,d^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fg,d^c=dg23 conjugacy classes — view the full character table.
| Fitting subgroup of | #2070⋊C3 |
| maximal subgroup of | C2×#2070, #23622, #24808, … |
| maximal normal subgroup of | C2×#2070, #23622, #24808, … |
| maximal quotient of | #10604, #10606, #10608, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 802);