Finite Group DatabasebetaFinite Group Databasebeta| order | 500 |
| isoclinism | isoclinic class of #92535 |
| automorphism | Aut(G) = #14522510 (order 2000) · Inn(G) = G · Out(G) = C4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C5 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽5) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, hypocenter | H(3,𝔽5) | C4 |
| Frattini, socle | C5 | #1042 |
| prime radical, cosocle | (C5⋊D5)⋊C5 | C2 |
| LCS | G ▷ H(3,𝔽5) | |
| UCS | C1 | |
23 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 25 |
A minimal faithful permutation representation — 2 generators on 25 points:
(2,4)(3,8,9,19)(5,11)(6,14,7,17)(10,18,20,16)(12,22,23,25)(13,21,15,24);(1,2,15,21,20)(3,7,5,24,4)(6,12,22,14,9)(8,18,13,25,10)(11,17,19,16,23)Polycyclic presentation:
2^2,5^3|a^2=b,c^a=c^4,d^a=d^3,e^a=e^2,d^b=d^4,e^b=e^4,d^c=de14 conjugacy classes — view the full character table.
| maximal subgroup of | (#1043⋊C5)⋊C2, C2×#92535, C3×#92535, … |
| maximal normal subgroup of | (#1043⋊C5)⋊C2, C2×#92535, C3×#92535 |
| maximal quotient of | #11758645, C2×#92535, C3×#92535, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(500, 21);