Finite Group DatabasebetaFinite Group Databasebeta| order | 128 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | Aut(G) = #72247039 (order 6144) · Inn(G) = C23×D4 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C23×D4 |
| commutator, Frattini, cosocle | C4 | C25 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups; only the 426 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 426 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,11)(4,8)(5,17)(6,9)(7,13)(10,19)(12,25)(14,24)(15,22)(16,29)(18,21)(20,27)(23,28)(26,32)(30,31);(1,3,6,14)(2,7,9,21)(4,12,15,26)(5,18,19,13)(8,20,22,31)(10,24,17,11)(16,30,28,27)(23,32,29,25);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,22)(10,23)(11,25)(13,27)(14,26)(17,29)(18,30)(19,28)(21,31)(24,32);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30);(2,9)(7,21)(8,22)(10,17)(11,24)(20,31)(23,29)(25,32)Polycyclic presentation:
2^7|b^2=g,f^2=g,b^a=bf,e^a=eg,f^a=fg,f^b=fg,d^c=dg41 conjugacy classes — view the full character table.
| Fitting subgroup of | #3586⋊C3, #11713197, #11713198, … |
| maximal subgroup of | #23873, #23875, #23884, … |
| maximal normal subgroup of | #23873, #23875, #23884, … |
| maximal quotient of | #60088, #60089, #60109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(128, 2318);