Finite Group DatabasebetaFinite Group Databasebeta| order | 43008 |
| isoclinism | isoclinic class of #72227554 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| Frattini | C25 | AGL(3,𝔽2) |
| Fitting, cosocle | C2×#2846 | GL(3,𝔽2) |
| socle | C23 | #72209612 |
| LCS | G | |
| UCS | C1 | |
508 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 9 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) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,64,50,10,33,18,16,63)(2,6,75,109,90,34,73,67)(3,35,57,48,91,36,55,119)(4,93,53,81,92,95,49,79)(5,123,115,60,87,38,66,82)(7,104,116,46,32,70,68,15)(8,76,117,14,31,122,69,47)(9,121,110,62,61,44,108,21)(11,19,59,83,65,126,23,37)(12,51,80,97,54,124,17,41)(13,30,96,128,107,89,94,125)(20,24,112,118,72,42,58,114)(22,27,102,101,86,28,105,77)(25,56,111,103,85,100,52,71)(26,98,120,78,40,127,43,88)(29,74,113,84,39,106,99,45);(2,3,11)(6,7,31)(9,35,42)(10,17,49)(12,18,50)(13,19,51)(14,62,43)(15,21,53)(16,55,73)(20,52,46)(22,81,110)(25,26,59)(27,87,39)(28,29,89)(32,34,33)(36,61,97)(37,94,66)(38,101,100)(40,85,92)(44,70,69)(45,71,114)(47,72,65)(48,82,78)(54,104,67)(56,74,115)(57,122,109)(58,76,117)(60,79,118)(63,91,111)(64,68,121)(75,116,112)(77,83,102)(80,120,90)(84,126,119)(86,88,124)(93,106,99)(95,123,125)(96,113,127)(98,105,128)(103,108,107)32 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((1,64,50,10,33,18,16,63)(2,6,75,109,90,34,73,67)(3,35,57,48,91,36,55,119)(4,93,53,81,92,95,49,79)(5,123,115,60,87,38,66,82)(7,104,116,46,32,70,68,15)(8,76,117,14,31,122,69,47)(9,121,110,62,61,44,108,21)(11,19,59,83,65,126,23,37)(12,51,80,97,54,124,17,41)(13,30,96,128,107,89,94,125)(20,24,112,118,72,42,58,114)(22,27,102,101,86,28,105,77)(25,56,111,103,85,100,52,71)(26,98,120,78,40,127,43,88)(29,74,113,84,39,106,99,45), (2,3,11)(6,7,31)(9,35,42)(10,17,49)(12,18,50)(13,19,51)(14,62,43)(15,21,53)(16,55,73)(20,52,46)(22,81,110)(25,26,59)(27,87,39)(28,29,89)(32,34,33)(36,61,97)(37,94,66)(38,101,100)(40,85,92)(44,70,69)(45,71,114)(47,72,65)(48,82,78)(54,104,67)(56,74,115)(57,122,109)(58,76,117)(60,79,118)(63,91,111)(64,68,121)(75,116,112)(77,83,102)(80,120,90)(84,126,119)(86,88,124)(93,106,99)(95,123,125)(96,113,127)(98,105,128)(103,108,107));