Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #72213660 |
| commutator, hypocenter | G | C1 |
| Frattini, Fitting, cosocle | #72267648 | A5 |
| socle | C25 | #14515303 |
| LCS | G | |
| UCS | C1 ◁ C2 | |
502 conjugacy classes of subgroups; only the 7 normal subgroups are shown.
This group has 502 conjugacy classes of subgroups — too many to draw. Only its 7 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,79,16,43,36)(2,38,39,32,62)(3,78,7,45,72)(4,75,5,20,60)(6,63,40,23,22)(8,48,46,37,74)(9,61,42,64,54)(10,65,17,57,56)(11,71,35,44,31)(12,80,30,49,47)(13,51,50,69,66)(14,77,24,52,55)(15,21,41,73,19)(18,26,59,70,25)(27,33,29,58,68)(28,76,34,67,53)(81,93,125,121,95,86,112,117,87,106)(82,116,91,126,124,101,97,115,111,89)(83,102)(84,114,107,110,90,98,119,128,108,99)(85,118,103,109,104,92,120,127,105,88)(94,100)(96,113)(122,123);(1,2,5,14,25,34)(3,9,7,11,19,35)(4,13,17)(6,18,36,42,61,72,46,38,55,21,15,31)(8,16,10)(12,23,24)(20,40,64,45,66,69,67,74,41,44,68,58)(22,47,50,33,60,26,48,56,29,51,53,39)(27,30,28)(32,57,73,37,52,70,59,49,54,63,43,62)(65,76,78,80,75,71)(77,79)(81,82,87,85,99,105,86,101,121,92,90,109)(83,91,111,94,119,107,102,115,126,100,114,128)(84,95,108,97,104,124,98,106,110,116,88,89)(93,117,123,120,103,96,112,125,122,118,127,113)52 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,79,16,43,36)(2,38,39,32,62)(3,78,7,45,72)(4,75,5,20,60)(6,63,40,23,22)(8,48,46,37,74)(9,61,42,64,54)(10,65,17,57,56)(11,71,35,44,31)(12,80,30,49,47)(13,51,50,69,66)(14,77,24,52,55)(15,21,41,73,19)(18,26,59,70,25)(27,33,29,58,68)(28,76,34,67,53)(81,93,125,121,95,86,112,117,87,106)(82,116,91,126,124,101,97,115,111,89)(83,102)(84,114,107,110,90,98,119,128,108,99)(85,118,103,109,104,92,120,127,105,88)(94,100)(96,113)(122,123), (1,2,5,14,25,34)(3,9,7,11,19,35)(4,13,17)(6,18,36,42,61,72,46,38,55,21,15,31)(8,16,10)(12,23,24)(20,40,64,45,66,69,67,74,41,44,68,58)(22,47,50,33,60,26,48,56,29,51,53,39)(27,30,28)(32,57,73,37,52,70,59,49,54,63,43,62)(65,76,78,80,75,71)(77,79)(81,82,87,85,99,105,86,101,121,92,90,109)(83,91,111,94,119,107,102,115,126,100,114,128)(84,95,108,97,104,124,98,106,110,116,88,89)(93,117,123,120,103,96,112,125,122,118,127,113));