Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92517 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | C125 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | C125 | C4 |
| Frattini | C25 | AGL(1,𝔽5) |
| socle | C5 | #1035 |
| prime radical, cosocle | D125 | C2 |
| LCS | G ▷ C125 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 125 |
A minimal faithful permutation representation — 2 generators on 125 points:
(2,5,6,16)(3,10,11,33)(4,14,15,45)(7,22,23,34)(8,26,27,17)(9,30,31,79)(12,39,40,44)(13,42,43,35)(18,52,53,87)(19,56,57,86)(20,58,37,92)(21,61,46,82)(24,66,67,60)(25,70,71,116)(28,75,63,113)(29,77,62,109)(32,84,85,48)(36,90,91,99)(38,41,47,97)(49,50,88,55)(51,78,54,106)(59,111,96,81)(64,95,93,108)(65,98,100,103)(68,110,115,114)(69,94,107,83)(72,117,118,89)(73,112,119,122)(74,80,101,121)(76,120,102,105)(104,125,124,123);(1,9,72,58,108,99,101,5,51,59,10,89,123,111,22,40,76,26,48,112,39,66,93,71,6,4,32,75,110,68,47,8,19,80,52,36,98,120,50,63,41,88,73,61,106,43,37,107,23,21,14,86,100,24,83,3,29,30,109,104,42,116,18,53,70,35,124,77,79,62,11,94,67,65,56,45,46,7,69,20,13,78,82,119,49,97,28,55,105,103,91,87,121,57,27,38,115,114,113,85,15,2,25,64,60,44,122,84,17,102,12,34,81,125,117,33,96,54,16,74,90,95,92,118,31)Polycyclic presentation:
2^2,5^3|a^2=b,c^5=de^3,d^5=e,c^a=c^2d,d^a=d^2e,e^a=e^2,c^b=c^4d^4e,d^b=d^4e^4,e^b=e^435 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#92517, C3×#92517, C3⋊#92517, … |
| maximal normal subgroup of | C2×#92517, C3×#92517, #72229806 |
| maximal quotient of | #11758618, C2×#92517, C3×#92517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 3);