Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | Aut(G) = #72267100 (order 12288) · Inn(G) = C22×C22⋊C4 · Out(G) = D4×S4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×C22⋊C4 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,18,23)(3,11,13,24,14,7,17,20)(4,8,16,22,15,21,28,31)(12,25,27,32,26,19,29,30);(1,3,6,14)(2,7,9,11)(4,12,15,26)(5,17,18,13)(8,19,21,25)(10,24,23,20)(16,29,28,27)(22,32,31,30);(1,4)(2,8)(3,12)(5,16)(6,15)(7,19)(9,21)(10,22)(11,25)(13,27)(14,26)(17,29)(18,28)(20,30)(23,31)(24,32);(4,15)(8,21)(12,26)(16,28)(19,25)(22,31)(27,29)(30,32)Polycyclic presentation:
2^7|a^2=f,b^2=g,f^2=g,b^a=be,e^a=eg,f^b=fg,d^c=dg41 conjugacy classes — view the full character table.
| Fitting subgroup of | #2889⋊C3, #11711618, #11711620, … |
| maximal subgroup of | #13096, #13098, #21780, … |
| maximal normal subgroup of | #13096, #13098, #21780, … |
| maximal quotient of | #17351, #17352, #17368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 1621);