Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | Aut(G) = #499787 (order 512) · Inn(G) = B(4,𝔽2) · Out(G) = D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C23 |
| prime radical | H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
113 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,28)(14,27)(15,26)(16,22)(17,23)(18,25)(19,29)(20,30)(21,31)(24,32);(1,3,6,14)(2,7,10,20)(4,15,17,12)(5,13,18,30)(8,21,23,19)(9,11,25,27)(16,31,32,29)(22,26,24,28);(1,4,6,17)(2,8,10,23)(3,12,14,15)(5,16,18,32)(7,19,20,21)(9,24,25,22)(11,26,27,28)(13,29,30,31)Polycyclic presentation:
2^7|b^2=g,c^2=g,b^a=bd,c^a=ce,f^a=fg,c^b=cg,e^b=efg,d^c=df,e^d=eg17 conjugacy classes — view the full character table.
| maximal subgroup of | #13679, #13680, C2×#2201, … |
| maximal normal subgroup of | #13679, #13680, C2×#2201, … |
| maximal quotient of | #12095, #12101, #12125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 933);